From cd7f8f85f2feb4c4e32908ea087d23d87226ce9b Mon Sep 17 00:00:00 2001
From: m-iancu <mihnea.iancu88@gmail.com>
Date: Wed, 6 Jul 2016 17:27:45 +0200
Subject: [PATCH] new source files
---
...oc Example) - Chapter 2: Other Markup.html | 482 ---
...r 9: The basic theory behind Wedderga.html | 860 -----
...ref) - Chapter 18: Cyclotomic Numbers.html | 1114 ------
...) - Chapter 60: Abelian Number Fields.html | 578 ---
... (ref) - Chapter 71: Character Tables.html | 3179 -----------------
...gap-system.org_Manuals_doc_ref_chap60.html | 472 +++
...anuals_pkg_GAPDoc-1.5.1_example_chap2.html | 373 ++
7 files changed, 845 insertions(+), 6213 deletions(-)
delete mode 100644 export/fromwebpage/GAP (GAPDoc Example) - Chapter 2: Other Markup.html
delete mode 100644 export/fromwebpage/GAP (Wedderga) - Chapter 9: The basic theory behind Wedderga.html
delete mode 100644 export/fromwebpage/GAP (ref) - Chapter 18: Cyclotomic Numbers.html
delete mode 100644 export/fromwebpage/GAP (ref) - Chapter 60: Abelian Number Fields.html
delete mode 100644 export/fromwebpage/GAP (ref) - Chapter 71: Character Tables.html
create mode 100644 export/fromwebpage/http:_www.gap-system.org_Manuals_doc_ref_chap60.html
create mode 100644 export/fromwebpage/http:_www.gap-system.org_Manuals_pkg_GAPDoc-1.5.1_example_chap2.html
diff --git a/export/fromwebpage/GAP (GAPDoc Example) - Chapter 2: Other Markup.html b/export/fromwebpage/GAP (GAPDoc Example) - Chapter 2: Other Markup.html
deleted file mode 100644
index 23096bf..0000000
--- a/export/fromwebpage/GAP (GAPDoc Example) - Chapter 2: Other Markup.html
+++ /dev/null
@@ -1,482 +0,0 @@
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-<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap0_mj.html">Top</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html">1</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html">2</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapA_mj.html">A</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html">B</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapBib_mj.html">Bib</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapInd_mj.html">Ind</a> </div>
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-<div class="chlinkprevnexttop"> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap0_mj.html">[Top of Book]</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap0_mj.html#contents">[Contents]</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html">[Previous Chapter]</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapA_mj.html">[Next Chapter]</a> </div>
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-<p id="mathjaxlink" class="pcenter"><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2.html">[MathJax off]</a><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chooser.html?BACK=http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html"> [Style]</a></p>
-<p><a id="X82793A7E7A3F2A54" name="X82793A7E7A3F2A54"></a></p>
-<div class="ChapSects"><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X82793A7E7A3F2A54">2 <span class="Heading">Other Markup</span></a>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X7A480B9A795EF264">2.1 <span class="Heading">Various types of text</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X7AA5BF0279938BE0">2.2 <span class="Heading">Formulae</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X833C410D85CF96A4">2.3 <span class="Heading">Crossreferencing</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X7F10E951789D6EDF">2.4 <span class="Heading">Lists and Tables</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X83A355E68485D6D1">2.5 <span class="Heading">Entities and Special Characters</span></a>
-</span>
-</div>
-</div>
-
-<h3>2 <span class="Heading">Other Markup</span></h3>
-
-<p><a id="X7A480B9A795EF264" name="X7A480B9A795EF264"></a></p>
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-<h4>2.1 <span class="Heading">Various types of text</span></h4>
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-<p>[→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X82A731CA83FB9ADD"><span class="RefLink">B.5</span></a>]</p>
-
-<p>In this section we present examples for all the various types of text that are possible in <strong class="pkg">GAPDoc</strong>:</p>
-
-
-<ul>
-<li><p><em>This</em> is <em>emphasized</em>.</p>
-
-</li>
-<li><p><em>Keywords</em> are typeset like <code class="keyw">this</code> and <code class="keyw">that</code>.</p>
-
-</li>
-<li><p><em>Arguments</em> of functions have an element. They look like this: <var class="Arg">x</var> and <var class="Arg">y</var>.</p>
-
-</li>
-<li><p><em>Code</em> can be written with the Code element: <code class="code">if x = y then Print("Equal"); fi;</code> or <code class="code">while true do Print("Hello"); od;</code>.</p>
-
-</li>
-<li><p><em>Filenames</em> have their own element: <code class="file">/usr/local/ca/gap4r2</code> or <code class="file">pkg/xgap/doc</code>.</p>
-
-</li>
-<li><p><em>Buttons</em>, <em>menus</em>, <em>menu entries</em>, and such things are also supported: <strong class="button">OK</strong> or <strong class="button">Cancel</strong>.</p>
-
-</li>
-<li><p><em>Packages</em> are typeset like this: <strong class="pkg">Small Groups Library</strong></p>
-
-</li>
-<li><p><em>Quoted</em> text: "This is a text in quotes."</p>
-
-</li>
-</ul>
-<p><em>Paragraphs</em> are separated by the empty <code class="code">Par</code> or <code class="code">P</code> element.</p>
-
-<p><em>Alternatives</em> for different output formats: This is other than LaTeX output, namely: <b>HTML</b> output.</p>
-
-<p>There are also three elements to typeset "verbatim-like" text. (→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X824BD70087820CF0"><span class="RefLink">B.6</span></a>)</p>
-
-<p>The first is a <em>Listing</em>:</p>
-
-
-<div class="example"><pre>Sieve := function(n)
- # Returns the primes less than n
- local l,p,i;
- l := [1..n]; Unbind(l[1]);
- p := 2;
- while p^2 <= n do
- if IsBound(l[p]) then
- i := 2 * p;
- while i <= n do Unbind(l[i]); i := i + p; od;
- fi;
- p := p + 1;
- od;
- return Compacted(l);
-end;
-</pre></div>
-
-<p>Here is a <em>Log</em> of a <strong class="pkg">GAP</strong> session using this function:</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">Sieve(100);</span>
-[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
- 67, 71, 73, 79, 83, 89, 97 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Length(last);</span>
-25
-</pre></div>
-
-<p>Here is a <strong class="pkg">GAP</strong> <em>Example</em> session that is automatically tested:</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">s := Size(CharacterTable("M"));</span>
-808017424794512875886459904961710757005754368000000000
-<span class="GAPprompt">gap></span> <span class="GAPinput">s < 10^53; </span>
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">s < 10^54;</span>
-true
-</pre></div>
-
-<p><a id="X7AA5BF0279938BE0" name="X7AA5BF0279938BE0"></a></p>
-
-<h4>2.2 <span class="Heading">Formulae</span></h4>
-
-<p>[→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X8516FAA07A95BBB5"><span class="RefLink">B.7</span></a>]</p>
-
-<p>There are three types of formulae.</p>
-
-<p>The first is the <em>normal math mode</em> of LaTeX: <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub><mo>&#x22C5;</mo><msub><mi>b</mi><mi>j</mi></msub><mo>=</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><msub><mi>h</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>b</mi><mi>k</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-1-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 10.641em; display: inline-block;" role="math" id="MathJax-Span-1" class="math"><span style="display: inline-block; position: relative; width: 8.173em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.529em, 1008.17em, 3.093em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2" class="mrow"><span id="MathJax-Span-3" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-5" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-6" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-7" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-8" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-9" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-10" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-11" class="munderover"><span style="display: inline-block; position: relative; width: 2.153em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-12" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.45em, 4.144em, -1000em); top: -4.478em; left: 0.914em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-13" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1001.24em, 4.142em, -1000em); top: -3.694em; left: 0.914em;"><span id="MathJax-Span-14" class="texatom"><span id="MathJax-Span-15" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-16" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-17" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-18" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-19" class="msubsup"><span style="display: inline-block; position: relative; width: 1.295em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.48em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-20" class="mi">h</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-21" class="texatom"><span id="MathJax-Span-22" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-23" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-24" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-25" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-26" class="msubsup"><span style="display: inline-block; position: relative; width: 0.901em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-27" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-28" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.458em; border-left: 0px solid; width: 0px; height: 1.783em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub><mo>⋅</mo><msub><mi>b</mi><mi>j</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><msub><mi>h</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>b</mi><mi>k</mi></msub></math></span></span><script id="MathJax-Element-1" type="math/tex">b_i \cdot b_j = \sum_{k=1}^d h_{ijk} b_k</script></span>. Then there are <em>displayed formulae</em>:</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"></span><div style="text-align: center;" class="MathJax_Display"><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">&#x27F9;</mo><mspace width="1em" /><mrow><mo>(</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><msub><mi>b</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>&#x22C5;</mo><mrow><mo>(</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><msub><mi>y</mi><mi>j</mi></msub><msub><mi>b</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><mrow><mo>(</mo><munder><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>x</mi><mi>i</mi></msub><msub><mi>y</mi><mi>j</mi></msub><msub><mi>h</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>)</mo></mrow><msub><mi>b</mi><mi>k</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-2-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 29.578em; display: inline-block;" role="math" id="MathJax-Span-29" class="math"><span style="display: inline-block; position: relative; width: 22.74em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.251em, 1022.74em, 5.723em, -1000em); top: -4.183em; left: 0em;"><span id="MathJax-Span-30" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-31" class="mo">⟹</span><span style="height: 0em; vertical-align: 0em; width: 1.125em; display: inline-block; overflow: hidden;" id="MathJax-Span-32" class="mspace"></span><span style="padding-left: 0.313em;" id="MathJax-Span-33" class="mrow"><span style="vertical-align: -0.778em;" id="MathJax-Span-34" class="mo"><span style="font-family: STIXSizeFourSym;">(</span></span><span id="MathJax-Span-35" class="munderover"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-36" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.96em, 4.242em, -1000em); top: -2.84em; left: 0.129em;"><span id="MathJax-Span-37" class="texatom"><span id="MathJax-Span-38" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-39" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-40" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-41" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.263em, 1000.37em, 4.144em, -1000em); top: -5.19em; left: 0.46em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-42" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-43" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-44" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-45" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-46" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-47" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-48" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.778em;" id="MathJax-Span-49" class="mo"><span style="font-family: STIXSizeFourSym;">)</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-50" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-51" class="mrow"><span style="vertical-align: -0.778em;" id="MathJax-Span-52" class="mo"><span style="font-family: STIXSizeFourSym;">(</span></span><span id="MathJax-Span-53" class="munderover"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-54" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.96em, 4.381em, -1000em); top: -2.84em; left: 0.128em;"><span id="MathJax-Span-55" class="texatom"><span id="MathJax-Span-56" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-57" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-58" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-59" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.263em, 1000.37em, 4.144em, -1000em); top: -5.19em; left: 0.46em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-60" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-61" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.43em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-62" class="mi">y</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-63" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-64" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-65" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-66" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.778em;" id="MathJax-Span-67" class="mo"><span style="font-family: STIXSizeFourSym;">)</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-68" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-69" class="munderover"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-70" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1001.09em, 4.242em, -1000em); top: -2.84em; left: 0.064em;"><span id="MathJax-Span-71" class="texatom"><span id="MathJax-Span-72" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-73" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-74" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-75" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.263em, 1000.37em, 4.144em, -1000em); top: -5.19em; left: 0.46em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-76" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-77" class="mrow"><span style="vertical-align: -0.778em;" id="MathJax-Span-78" class="mo"><span style="font-family: STIXSizeFourSym;">(</span></span><span id="MathJax-Span-79" class="munderover"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-80" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.57em, 4.381em, -1000em); top: -2.84em; left: 0.361em;"><span id="MathJax-Span-81" class="texatom"><span id="MathJax-Span-82" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-83" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-84" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-85" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-86" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-87" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-88" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-89" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.43em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-90" class="mi">y</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-91" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-92" class="msubsup"><span style="display: inline-block; position: relative; width: 1.295em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.48em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-93" class="mi">h</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-94" class="texatom"><span id="MathJax-Span-95" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-96" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-97" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-98" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.778em;" id="MathJax-Span-99" class="mo"><span style="font-family: STIXSizeFourSym;">)</span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-100" class="msubsup"><span style="display: inline-block; position: relative; width: 0.901em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-101" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-102" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.183em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.878em; border-left: 0px solid; width: 0px; height: 4.264em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML MJX_Assistive_MathML_Block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">⟹</mo><mspace width="1em"></mspace><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><msub><mi>b</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><msub><mi>y</mi><mi>j</mi></msub><msub><mi>b</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><mrow><mo>(</mo><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>x</mi><mi>i</mi></msub><msub><mi>y</mi><mi>j</mi></msub><msub><mi>h</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>)</mo></mrow><msub><mi>b</mi><mi>k</mi></msub></math></span></span></div><script id="MathJax-Element-2" type="math/tex; mode=display">
- \Longrightarrow \quad \left(\sum_{i=1}^d x_i b_i \right) \cdot
- \left(\sum_{j=1}^d y_j b_j \right) =
- \sum_{k=1}^d \left( \sum_{i,j} x_i y_j h_{ijk} \right) b_k
-</script></p>
-
-<p>If possible, use the <code class="code">Alt</code> element to specify a better readable text version of such a formula as in the following example:</p>
-
-
-<pre class="normal">
- d d d
- ----- ----- ----- -----
- \ \ \ \
- ==> ( ) x_i b_i )( ) y_j b_j ) = ) ( ) x_i y_j h_ijk ) b_k
- / / / /
- ----- ----- ----- -----
- i = 1 j = 1 k = 1 i,j
-
-</pre>
-
-<p>For small formulae without "difficult" parts use the <code class="code">M</code> element: <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-3-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-103" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.77em, 2.754em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-104" class="mrow"><span id="MathJax-Span-105" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-106" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-107" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub></math></span></span><script id="MathJax-Element-3" type="math/tex">b_i</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-4-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.203em; display: inline-block;" role="math" id="MathJax-Span-108" class="math"><span style="display: inline-block; position: relative; width: 0.913em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.467em, 1000.91em, 2.607em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-109" class="mrow"><span id="MathJax-Span-110" class="msubsup"><span style="display: inline-block; position: relative; width: 0.926em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-111" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.498em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-112" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.232em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math></span></span><script id="MathJax-Element-4" type="math/tex">x^2</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-5-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 11.891em; display: inline-block;" role="math" id="MathJax-Span-113" class="math"><span style="display: inline-block; position: relative; width: 9.135em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.659em, 1009.14em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-114" class="mrow"><span id="MathJax-Span-115" class="msubsup"><span style="display: inline-block; position: relative; width: 0.926em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-116" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.498em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-117" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-118" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-119" class="mn">2</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-120" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-121" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-122" class="mn">1</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-123" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-124" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-125" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-126" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-127" class="mn">1</span><span id="MathJax-Span-128" class="msubsup"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-129" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-130" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.448em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></span></span><script id="MathJax-Element-5" type="math/tex">x^2 + 2x + 1 = (x + 1)^2</script></span>. Note that here whitespace matters for text (or HTML) output.</p>
-
-<p>Here are two formulae containing less than characters which are special characters for XML: <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>&lt;</mo><mi>b</mi><mo>&lt;</mo><mi>c</mi><mo>&lt;</mo><mi>d</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-6-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.578em; display: inline-block;" role="math" id="MathJax-Span-131" class="math"><span style="display: inline-block; position: relative; width: 5.817em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1005.82em, 2.812em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-132" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-133" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-134" class="mo"><</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-135" class="mi">b</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-136" class="mo"><</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-137" class="mi">c</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-138" class="mo"><</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-139" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.094em; border-left: 0px solid; width: 0px; height: 1.044em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo><</mo><mi>b</mi><mo><</mo><mi>c</mi><mo><</mo><mi>d</mi></math></span></span><script id="MathJax-Element-6" type="math/tex">a < b < c < d</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>&lt;</mo><mi>f</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-7-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.828em; display: inline-block;" role="math" id="MathJax-Span-140" class="math"><span style="display: inline-block; position: relative; width: 2.163em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.822em, 1002.16em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-141" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-142" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-143" class="mo"><</span><span style="font-family: STIXGeneral; font-style: italic; text-rendering: optimizelegibility; padding-left: 0.313em;" id="MathJax-Span-144" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.276em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo><</mo><mi>f</mi></math></span></span><script id="MathJax-Element-7" type="math/tex">e < f</script></span>.</p>
-
-<p>Using the <code class="code">Mode</code> attribute of a <code class="code">Display</code> element formulae like</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"></span><div style="text-align: center;" class="MathJax_Display"><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>a</mi><mo stretchy="false">&#x27F6;</mo><mi>a</mi><mo lspace="thickmathspace" rspace="thickmathspace">mod</mo><mi>m</mi><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-8-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.578em; display: inline-block;" role="math" id="MathJax-Span-145" class="math"><span style="display: inline-block; position: relative; width: 6.587em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1006.54em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-146" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-147" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-148" class="mo">⟶</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-149" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.313em; padding-right: 0.313em;" id="MathJax-Span-150" class="mo">mod</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-151" class="mi">m</span><span style="font-family: STIXVariants;" id="MathJax-Span-152" class="mi">′</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML MJX_Assistive_MathML_Block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>a</mi><mo stretchy="false">⟶</mo><mi>a</mi><mo lspace="thickmathspace" rspace="thickmathspace">mod</mo><mi>m</mi><mi class="MJX-variant" mathvariant="normal">′</mi></math></span></span></div><script id="MathJax-Element-8" type="math/tex; mode=display">a \longrightarrow a \bmod m\prime</script></p>
-
-<p>can also be displayed nicely in text and HTML output.</p>
-
-<p><a id="X833C410D85CF96A4" name="X833C410D85CF96A4"></a></p>
-
-<h4>2.3 <span class="Heading">Crossreferencing</span></h4>
-
-<p>[→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X7D19CF4782309661"><span class="RefLink">B.8</span></a>]</p>
-
-<p>In this section we demonstrate various references to parts of this document. Here is a reference to this section: <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">2.3</span></a>. Here is a reference to chapter <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a>, to appendix <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapA_mj.html#X7B53252784137533"><span class="RefLink">A</span></a>, and to subsection <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7E193BD379F58A4C"><span class="RefLink">1.1-1</span></a>.</p>
-
-<p>We distinguish among others references to functions (see <code class="func">f</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7FA1D0937FA1D093"><span class="RefLink">1.2-1</span></a>)), to methods with tricky name (see <code class="func">\^\{\}\[\]\<\&</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X822B5C487B29E799"><span class="RefLink">1.2-2</span></a>)), to operations (see <code class="func">MyOperation</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7D33C2597988F481"><span class="RefLink">1.2-3</span></a>)), to methods (see <code class="func">MyOperation</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X783DCD4E826289D4"><span class="RefLink">1.2-4</span></a>) or <code class="func">MyOperation</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7A5F4A287D06988C"><span class="RefLink">1.2-5</span></a>)), to filters (see <code class="func">IsBla</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X82954B687D2DF3C2"><span class="RefLink">1.2-6</span></a>)), to properties (see <code class="func">IsBlubb</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X80C364DD7C919CCE"><span class="RefLink">1.2-7</span></a>)), to attributes (see <code class="func">NumberBlobbs</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X8052A45E7F9F054C"><span class="RefLink">1.2-8</span></a>)), to variables (<code class="func">AllBlibbs</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7C00E05A7DDEF003"><span class="RefLink">1.2-9</span></a>)), to families (see <code class="func">BlibbsFamily</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7CBC935A8142E374"><span class="RefLink">1.2-10</span></a>)), and to info classes (see <code class="func">InfoBlibbs</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X84D7D77378AD030A"><span class="RefLink">1.2-11</span></a>)).</p>
-
-<p>There are also references to labels: see <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">here</span></a>, to other books: see <a href="http://www.gap-system.org/Manuals/pkg/pkg/GAPDoc-1.5.1/doc/chap3_mj.html#X7B76F6F786521F6B"><span class="RefLink">GAPDoc: What is a DTD?</span></a> or <code class="func">IsSubgroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap40_mj.html#X7839D8927E778334"><span class="RefLink">Reference: IsSubgroup</span></a>) in the <strong class="pkg">GAP</strong> reference manual.</p>
-
-<p>References to sections come in two styles: <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a> or <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink"><span class="Heading">Sectioning Elements</span></span></a>.</p>
-
-<p>Another type of cross referencing is bibliography. Here is a citation: <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapBib_mj.html#biBCR1">[CR81, (5.22)]</a> is an interesting lemma.</p>
-
-<p>There are also URLs:</p>
-
-<p><span class="URL"><a href="http://www.math.rwth-aachen.de/">http://www.math.rwth-aachen.de/</a></span></p>
-
-<p>Email addresses have a special element: <span class="URL"><a href="mailto:Frank.Luebeck@Math.RWTH-Aachen.De">Frank.Luebeck@Math.RWTH-Aachen.De</a></span></p>
-
-<p>and Homepages another one: <span class="URL"><a href="http://www-groups.mcs.st-and.ac.uk/%7Eneunhoef/">http://www-groups.mcs.st-and.ac.uk/~neunhoef/</a></span></p>
-
-<p>And here is a link to the <span class="URL"><a href="http://www.math.rwth-aachen.de/%7EFrank.Luebeck/gap/EDIM/index.html#ARCHS"><strong class="pkg">EDIM</strong> archives</a></span>.</p>
-
-<p>One can generate index entries as follows (look up the words "TeX-UserGroup", "RWTH", "Aachen, Hauptbahnhof", and "<strong class="pkg">GAP</strong>, <strong class="pkg">GAPDoc</strong>").</p>
-
-<p><a id="X7F10E951789D6EDF" name="X7F10E951789D6EDF"></a></p>
-
-<h4>2.4 <span class="Heading">Lists and Tables</span></h4>
-
-<p>[→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X7BB822947F626E1A"><span class="RefLink">B.9</span></a>]</p>
-
-<p>There are</p>
-
-
-<ul>
-<li><p>lists</p>
-
-</li>
-<li><p>enumerations, and</p>
-
-</li>
-<li><p>tables</p>
-
-</li>
-</ul>
-<p>or:</p>
-
-<ol>
-<li><p>lists</p>
-
-</li>
-<li><p>enumerations, and</p>
-
-</li>
-<li><p>tables</p>
-
-</li>
-</ol>
-<p>or with marks:</p>
-
-
-<dl>
-<dt><strong class="Mark">lists:</strong></dt>
-<dd><p>not numbered</p>
-
-</dd>
-<dt><strong class="Mark">enumerations:</strong></dt>
-<dd><p>numbered</p>
-
-</dd>
-<dt><strong class="Mark">tables:</strong></dt>
-<dd><p>two-dimensional</p>
-
-</dd>
-</dl>
-<p>Lists can also be nested:</p>
-
-<ol>
-<li><ol>
-<li><p>first item of inner enumeration</p>
-
-</li>
-<li><p>second item of inner enumeration</p>
-
-</li>
-</ol>
-</li>
-<li>
-<ul>
-<li><p>first item of inner list</p>
-
-</li>
-<li><p>second item of inner list</p>
-
-</li>
-</ul>
-</li>
-</ol>
-<p>Here is a <em>table</em>:</p>
-
-<div class="pcenter"><table class="GAPDocTable">
-<caption class="GAPDocTable"><b>Table: </b>Prices</caption>
-<tbody><tr>
-<td class="tdright">Object</td>
-<td class="tdcenter">Price</td>
-<td class="tdleft">available</td>
-</tr>
-<tr>
-<td class="tdright">Shoe</td>
-<td class="tdcenter">$1,00</td>
-<td class="tdleft">there</td>
-</tr>
-<tr>
-<td class="tdright">Hat</td>
-<td class="tdcenter">$2,00</td>
-<td class="tdleft">not there</td>
-</tr>
-</tbody></table><br><p> </p><br>
-</div>
-
-<p><a id="X83A355E68485D6D1" name="X83A355E68485D6D1"></a></p>
-
-<h4>2.5 <span class="Heading">Entities and Special Characters</span></h4>
-
-<p>[→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X80B478CD7E584F6F"><span class="RefLink">B.10</span></a>]</p>
-
-<p>Here is a table of special characters, the first two are special for XML and must be typed in by entities in <strong class="pkg">GAPDoc</strong> documents. The other characters are special for LaTeX but in <strong class="pkg">GAPDoc</strong> they can be typed directly.</p>
-
-<div class="pcenter"><table class="GAPDocTable">
-<caption class="GAPDocTable"><b>Table: </b>Special characters in character data</caption>
-<tbody><tr>
-<td class="tdcenter"><code class="code">&</code></td>
-<td class="tdcenter"><code class="code"><</code></td>
-<td class="tdcenter"><code class="code">></code></td>
-<td class="tdcenter"><code class="code">#</code></td>
-<td class="tdcenter"><code class="code">$</code></td>
-<td class="tdcenter"><code class="code">%</code></td>
-<td class="tdcenter"><code class="code">~</code></td>
-<td class="tdcenter"><code class="code">\</code></td>
-<td class="tdcenter"><code class="code">{</code></td>
-<td class="tdcenter"><code class="code">}</code></td>
-<td class="tdcenter"><code class="code">_</code></td>
-<td class="tdcenter"><code class="code">^</code></td>
-<td class="tdcenter"><code class="code"> </code></td>
-</tr>
-</tbody></table><br><p> </p><br>
-</div>
-
-<p>And here are the predefined entities in <strong class="pkg">GAPDoc</strong>:</p>
-
-<div class="pcenter"><table class="GAPDocTable">
-<caption class="GAPDocTable"><b>Table: </b>Predefined Entities in the <strong class="pkg">GAPDoc</strong> system</caption>
-<tbody><tr>
-<td class="tdleft"><code class="code">&GAP;</code></td>
-<td class="tdleft"><strong class="pkg">GAP</strong></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">&GAPDoc;</code></td>
-<td class="tdleft"><strong class="pkg">GAPDoc</strong></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">&TeX;</code></td>
-<td class="tdleft">TeX</td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">&LaTeX;</code></td>
-<td class="tdleft">LaTeX</td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">&BibTeX;</code></td>
-<td class="tdleft">BibTeX</td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">&MeatAxe;</code></td>
-<td class="tdleft"><strong class="pkg">MeatAxe</strong></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">&XGAP;</code></td>
-<td class="tdleft"><strong class="pkg">XGAP</strong></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">&copyright;</code></td>
-<td class="tdleft">©</td>
-</tr>
-</tbody></table><br><p> </p><br>
-</div>
-
-<p>And some more for mathematical symbols: â„‚, ℤ, â„•, â„™, â„š, â„, â„.</p>
-
-
-<div class="chlinkprevnextbot"> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap0_mj.html">[Top of Book]</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap0_mj.html#contents">[Contents]</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html">[Previous Chapter]</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapA_mj.html">[Next Chapter]</a> </div>
-
-
-<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap0_mj.html">Top</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html">1</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html">2</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapA_mj.html">A</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html">B</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapBib_mj.html">Bib</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapInd_mj.html">Ind</a> </div>
-
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-<div class="ChapSects"><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8487E7AD7D6B2EA5">9 <span class="Heading">The basic theory behind <strong class="pkg">Wedderga</strong></span></a>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X815ECCD97B18314B">9.1 <span class="Heading">Group rings and group algebras</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7FDD93FB79ADCC91">9.2 <span class="Heading">Semisimple group algebras</span></a>
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-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84BB4A6081EAE905">9.3 <span class="Heading">Wedderburn components</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X87B6505C7C2EE054">9.4 <span class="Heading">Characters and primitive central idempotents</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7A24D5407F72C633">9.5 <span class="Heading">Central simple algebras and Brauer equivalence</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7FB21779832CE1CB">9.6 <span class="Heading">Crossed Products</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X828C42CD86AF605F">9.7 <span class="Heading">Cyclic Crossed Products</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7869E2A48784C232">9.8 <span class="Heading">Abelian Crossed Products</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X80BABE5078A29793">9.9 <span class="Heading">Classical crossed products</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84C98BB8859BBEE2">9.10 <span class="Heading">Cyclic Algebras</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8099A8C784255672">9.11 <span class="Heading">Cyclotomic algebras</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8310E96086509397">9.13 <span class="Heading">Idempotents given by subgroups</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7D518BAB80EDE190">9.14 <span class="Heading">Shoda pairs of a group</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7E3479527BAE5B9E">9.15 <span class="Heading">Strong Shoda pairs of a group</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84C694978557EFE5">9.16 <span class="Heading">Strongly monomial characters and strongly monomial groups</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X800D8C5087D79DC8">9.17 <span class="Heading">Cyclotomic Classes and Strong Shoda Pairs</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X803562E087325AF6">9.18 <span class="Heading">Theory for Local Schur Index and Division Algebra Part Calculations</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8472ACCF802EC188">9.20 <span class="Heading">A complete set of orthogonal primitive idempotents</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X856D7975810BF987">9.21 <span class="Heading">Applications to coding theory</span></a>
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-<h3>9 <span class="Heading">The basic theory behind <strong class="pkg">Wedderga</strong></span></h3>
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-<p>In this chapter we describe the theory that is behind the algorithms used by <strong class="pkg">Wedderga</strong>.</p>
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-<p>All the rings considered in this chapter are associative and have an identity.</p>
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-<p>We use the following notation: <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1" class="MJXp-math"><span id="MJXp-Span-2" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3" class="MJXp-mo">â„š</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-1-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2" class="mrow"><span id="MathJax-Span-3" class="texatom"><span id="MathJax-Span-4" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-5" class="mo">â„š</span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-1" type="math/tex">â„š</script></span> denotes the field of rationals and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4" class="MJXp-math"><span id="MJXp-Span-5" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-6" class="MJXp-mrow"><span id="MJXp-Span-7" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-8" class="MJXp-mi MJXp-italic MJXp-script">q</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-2-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-6" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-7" class="mrow"><span id="MathJax-Span-8" class="msubsup"><span style="display: inline-block; position: relative; width: 0.898em; height: 0px;"><span style="position: absolute; clip: rect(3.184em, 1000.57em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-9" class="texatom"><span id="MathJax-Span-10" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-11" class="mi">ð”½<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.098em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.469em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-12" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-2" type="math/tex">\mathbb F_q</script></span> the finite field of order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-9" class="MJXp-math"><span id="MJXp-Span-10" class="MJXp-mi MJXp-italic">q</span></span></span><span style="" tabindex="0" id="MathJax-Element-3-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-13" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-14" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-15" class="mi">q</span></span></span></span></span></nobr></span><script id="MathJax-Element-3" type="math/tex">q</script></span>. For every positive integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-11" class="MJXp-math"><span id="MJXp-Span-12" class="MJXp-mi MJXp-italic">k</span></span></span><span style="" tabindex="0" id="MathJax-Element-4-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-16" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-17" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-18" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-4" type="math/tex">k</script></span>, we denote a complex <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-13" class="MJXp-math"><span id="MJXp-Span-14" class="MJXp-mi MJXp-italic">k</span></span></span><span style="" tabindex="0" id="MathJax-Element-5-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-19" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-20" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-21" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-5" type="math/tex">k</script></span>-th primitive root of unity by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-15" class="MJXp-math"><span id="MJXp-Span-16" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-17" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-18" class="MJXp-mi MJXp-italic MJXp-script">k</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-6-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-22" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-23" class="mrow"><span id="MathJax-Span-24" class="msubsup"><span style="display: inline-block; position: relative; width: 0.855em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-25" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.454em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-26" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-6" type="math/tex">\xi_k</script></span> and so <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-19" class="MJXp-math"><span id="MJXp-Span-20" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-21" class="MJXp-mo">â„š</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-22" class="MJXp-mo">(</span><span id="MJXp-Span-23" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-24" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-25" class="MJXp-mi MJXp-italic MJXp-script">k</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-26" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-7-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-27" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-28" class="mrow"><span id="MathJax-Span-29" class="texatom"><span id="MathJax-Span-30" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-31" class="mo">â„š</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-32" class="mo">(</span><span id="MathJax-Span-33" class="msubsup"><span style="display: inline-block; position: relative; width: 0.855em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-34" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.454em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-35" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-36" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-7" type="math/tex">â„š(\xi_k)</script></span> is the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-27" class="MJXp-math"><span id="MJXp-Span-28" class="MJXp-mi MJXp-italic">k</span></span></span><span style="" tabindex="0" id="MathJax-Element-8-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-37" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-38" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-39" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-8" type="math/tex">k</script></span>-th cyclotomic extension of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-29" class="MJXp-math"><span id="MJXp-Span-30" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-31" class="MJXp-mo">â„š</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-9-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-40" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-41" class="mrow"><span id="MathJax-Span-42" class="texatom"><span id="MathJax-Span-43" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-44" class="mo">â„š</span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-9" type="math/tex">â„š</script></span>.</p>
-
-<p><a id="X815ECCD97B18314B" name="X815ECCD97B18314B"></a></p>
-
-<h4>9.1 <span class="Heading">Group rings and group algebras</span></h4>
-
-<p>Given a group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-32" class="MJXp-math"><span id="MJXp-Span-33" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-10-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-45" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-46" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-47" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-10" type="math/tex">G</script></span> and a ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-34" class="MJXp-math"><span id="MJXp-Span-35" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-11-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-48" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-49" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-50" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-11" type="math/tex">R</script></span>, the <em>group ring</em> <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-36" class="MJXp-math"><span id="MJXp-Span-37" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-38" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-12-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-51" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-52" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-53" class="mi">R</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-54" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-12" type="math/tex">RG</script></span> over the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-39" class="MJXp-math"><span id="MJXp-Span-40" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-13-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-55" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-56" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-57" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-13" type="math/tex">G</script></span> with coefficients in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-41" class="MJXp-math"><span id="MJXp-Span-42" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-14-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-58" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-59" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-60" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-14" type="math/tex">R</script></span> is the ring whose underlying additive group is a right <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-43" class="MJXp-math"><span id="MJXp-Span-44" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-45" class="MJXp-mo">−</span></span></span><span style="" tabindex="0" id="MathJax-Element-15-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-61" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-62" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-63" class="mi">R</span><span style="font-family: STIXGeneral;" id="MathJax-Span-64" class="mo">−</span></span></span></span></span></nobr></span><script id="MathJax-Element-15" type="math/tex">R-</script></span>module with basis <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-46" class="MJXp-math"><span id="MJXp-Span-47" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-16-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-65" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-66" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-67" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-16" type="math/tex">G</script></span> such that the product is defined by the following rule</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-48" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-49" class="MJXp-mo">(</span><span id="MJXp-Span-50" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-51" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-52" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-53" class="MJXp-mo">(</span><span id="MJXp-Span-54" class="MJXp-mi MJXp-italic">h</span><span id="MJXp-Span-55" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-56" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-57" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-58" class="MJXp-mo">(</span><span id="MJXp-Span-59" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-60" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-61" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-62" class="MJXp-mo">(</span><span id="MJXp-Span-63" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-64" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-65" class="MJXp-mo">)</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-17-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-68" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-69" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-70" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-71" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-72" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-73" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-74" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-75" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-76" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-77" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-78" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-79" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-80" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-81" class="mi">h</span><span style="font-family: STIXGeneral;" id="MathJax-Span-82" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-83" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-84" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-85" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-86" class="mo">)</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-17" type="math/tex; mode=display">
- (gr)(hs)=(gh)(rs)
- </script></p>
-
-<p>for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-66" class="MJXp-math"><span id="MJXp-Span-67" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-68" class="MJXp-mo">,</span><span id="MJXp-Span-69" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-70" class="MJXp-mo">∈</span><span id="MJXp-Span-71" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-18-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-87" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-88" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-89" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-90" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-91" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-92" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-93" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-18" type="math/tex">r,s \in R</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-72" class="MJXp-math"><span id="MJXp-Span-73" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-74" class="MJXp-mo">,</span><span id="MJXp-Span-75" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-76" class="MJXp-mo">∈</span><span id="MJXp-Span-77" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-19-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-94" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-95" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-96" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-97" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-98" class="mi">h</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-99" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-100" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-19" type="math/tex">g, h \in G</script></span>, and extended to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-78" class="MJXp-math"><span id="MJXp-Span-79" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-80" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-20-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-101" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-102" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-103" class="mi">R</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-104" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-20" type="math/tex">RG</script></span> by linearity.</p>
-
-<p>A <em>group algebra</em> is a group ring in which the coefficient ring is a field.</p>
-
-<p><a id="X7FDD93FB79ADCC91" name="X7FDD93FB79ADCC91"></a></p>
-
-<h4>9.2 <span class="Heading">Semisimple group algebras</span></h4>
-
-<p>We say that a ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-81" class="MJXp-math"><span id="MJXp-Span-82" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-21-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-105" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-106" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-107" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-21" type="math/tex">R</script></span> is semisimple if it is a direct sum of simple left (alternatively right) ideals or equivalently if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-83" class="MJXp-math"><span id="MJXp-Span-84" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-22-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-108" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-109" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-110" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-22" type="math/tex">R</script></span> is isomorphic to a direct product of simple algebras each one isomorphic to a matrix ring over a division ring.</p>
-
-<p>By Maschke's Theorem, if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-85" class="MJXp-math"><span id="MJXp-Span-86" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-23-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-111" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-112" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-113" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-23" type="math/tex">G</script></span> is a finite group then the group algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-87" class="MJXp-math"><span id="MJXp-Span-88" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-89" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-24-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-114" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-115" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-116" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-117" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-24" type="math/tex">FG</script></span> is semisimple if and only the characteristic of the coefficient field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-90" class="MJXp-math"><span id="MJXp-Span-91" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-25-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-118" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-119" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-120" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-25" type="math/tex">F</script></span> does not divide the order of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-92" class="MJXp-math"><span id="MJXp-Span-93" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-26-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-121" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-122" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-123" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-26" type="math/tex">G</script></span>.</p>
-
-<p>In fact, an arbitrary group ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-94" class="MJXp-math"><span id="MJXp-Span-95" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-96" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-27-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-124" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-125" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-126" class="mi">R</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-127" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-27" type="math/tex">RG</script></span> is semisimple if and only if the coefficient ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-97" class="MJXp-math"><span id="MJXp-Span-98" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-28-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-128" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-129" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-130" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-28" type="math/tex">R</script></span> is semisimple, the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-99" class="MJXp-math"><span id="MJXp-Span-100" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-29-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-131" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-132" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-133" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-29" type="math/tex">G</script></span> is finite and the order of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-101" class="MJXp-math"><span id="MJXp-Span-102" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-30-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-134" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-135" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-136" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-30" type="math/tex">G</script></span> is invertible in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-103" class="MJXp-math"><span id="MJXp-Span-104" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-31-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-137" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-138" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-139" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-31" type="math/tex">R</script></span>.</p>
-
-<p>Some authors use the notion semisimple ring for rings with zero
-Jacobson radical. To avoid confusion we usually refer to semisimple
-rings as semisimple artinian rings.</p>
-
-<p><a id="X84BB4A6081EAE905" name="X84BB4A6081EAE905"></a></p>
-
-<h4>9.3 <span class="Heading">Wedderburn components</span></h4>
-
-<p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-105" class="MJXp-math"><span id="MJXp-Span-106" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-32-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-140" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-141" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-142" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-32" type="math/tex">R</script></span> is a <em>semisimple ring</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7FDD93FB79ADCC91"><span class="RefLink">9.2</span></a>) then the <em>Wedderburn decomposition</em> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-107" class="MJXp-math"><span id="MJXp-Span-108" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-33-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-143" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-144" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-145" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-33" type="math/tex">R</script></span> is the decomposition of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-109" class="MJXp-math"><span id="MJXp-Span-110" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-34-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-146" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-147" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-148" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-34" type="math/tex">R</script></span> as a direct product of simple algebras. The factors of this Wedderburn decomposition are called <em>Wedderburn components</em> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-111" class="MJXp-math"><span id="MJXp-Span-112" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-35-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-149" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-150" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-151" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-35" type="math/tex">R</script></span>. Each Wedderburn component of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-113" class="MJXp-math"><span id="MJXp-Span-114" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-36-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-152" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-153" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-154" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-36" type="math/tex">R</script></span> is of the form <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-115" class="MJXp-math"><span id="MJXp-Span-116" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-117" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-37-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-155" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-156" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-157" class="mi">R</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-158" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-37" type="math/tex">Re</script></span> for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-118" class="MJXp-math"><span id="MJXp-Span-119" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-38-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-159" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-160" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-161" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-38" type="math/tex">e</script></span> a <em>primitive central idempotent</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-120" class="MJXp-math"><span id="MJXp-Span-121" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-39-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-162" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-163" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-164" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-39" type="math/tex">R</script></span>.</p>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-122" class="MJXp-math"><span id="MJXp-Span-123" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-124" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-40-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-165" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-166" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-167" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-168" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-40" type="math/tex">FG</script></span> be a <em>semisimple group algebra</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7FDD93FB79ADCC91"><span class="RefLink">9.2</span></a>). If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-125" class="MJXp-math"><span id="MJXp-Span-126" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-41-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-169" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-170" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-171" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-41" type="math/tex">F</script></span> has positive characteristic, then the Wedderburn components of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-127" class="MJXp-math"><span id="MJXp-Span-128" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-129" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-42-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-172" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-173" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-174" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-175" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-42" type="math/tex">FG</script></span> are matrix algebras over finite extensions of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-130" class="MJXp-math"><span id="MJXp-Span-131" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-43-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-176" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-177" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-178" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-43" type="math/tex">F</script></span>. If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-132" class="MJXp-math"><span id="MJXp-Span-133" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-44-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-179" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-180" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-181" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-44" type="math/tex">F</script></span> has zero characteristic then by the <em>Brauer-Witt Theorem</em> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBY">[Yam74]</a>, the <em>Wedderburn components</em> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-134" class="MJXp-math"><span id="MJXp-Span-135" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-136" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-45-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-182" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-183" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-184" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-185" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-45" type="math/tex">FG</script></span> are <em>Brauer equivalent</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to <em>cyclotomic algebras</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>).</p>
-
-<p>The main functions of <strong class="pkg">Wedderga</strong> compute the Wedderburn components of a semisimple group algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-137" class="MJXp-math"><span id="MJXp-Span-138" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-139" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-46-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-186" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-187" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-188" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-189" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-46" type="math/tex">FG</script></span>,
- such that the coefficient field is either an abelian number field (i.e.
- a subfield of a finite cyclotomic extension of the rationals) or a
-finite field. In the finite case, the Wedderburn components are matrix
-algebras over finite fields and so can be described by the size of the
-matrices and the size of the finite field.</p>
-
-<p>In the zero characteristic case each Wedderburn component <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-140" class="MJXp-math"><span id="MJXp-Span-141" class="MJXp-mi MJXp-italic">A</span></span></span><span style="" tabindex="0" id="MathJax-Element-47-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-190" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-191" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-192" class="mi">A</span></span></span></span></span></nobr></span><script id="MathJax-Element-47" type="math/tex">A</script></span> is <em>Brauer equivalent</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to a <em>cyclotomic algebra</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>) and therefore <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-142" class="MJXp-math"><span id="MJXp-Span-143" class="MJXp-mi MJXp-italic">A</span></span></span><span style="" tabindex="0" id="MathJax-Element-48-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-193" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-194" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-195" class="mi">A</span></span></span></span></span></nobr></span><script id="MathJax-Element-48" type="math/tex">A</script></span> is a (possibly fractional) matrix algebra over <em>cyclotomic algebra</em> and can be described numerically in one of the following three forms:</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-144" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-145" class="MJXp-mo">[</span><span id="MJXp-Span-146" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-147" class="MJXp-mo">,</span><span id="MJXp-Span-148" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-149" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-150" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-49-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-196" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-197" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-198" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-199" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-200" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-201" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-202" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-203" class="mo">,</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-49" type="math/tex; mode=display">
- [n,K],
- </script></p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-151" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-152" class="MJXp-mo">[</span><span id="MJXp-Span-153" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-154" class="MJXp-mo">,</span><span id="MJXp-Span-155" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-156" class="MJXp-mo">,</span><span id="MJXp-Span-157" class="MJXp-mi MJXp-italic">k</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-158" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-159" class="MJXp-mo">[</span><span id="MJXp-Span-160" class="MJXp-mi MJXp-italic">d</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-161" class="MJXp-mo">,</span><span id="MJXp-Span-162" class="MJXp-mi MJXp-italic">α</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-163" class="MJXp-mo">,</span><span id="MJXp-Span-164" class="MJXp-mi MJXp-italic">β</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-165" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-166" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-167" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-50-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-204" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-205" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-206" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-207" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-208" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-209" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-210" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-211" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-212" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-213" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-214" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-215" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-216" class="mi">α</span><span style="font-family: STIXGeneral;" id="MathJax-Span-217" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-218" class="mi">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-219" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-220" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-221" class="mo">,</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-50" type="math/tex; mode=display">
- [n,K,k,[d,\alpha,\beta]],
- </script></p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-168" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-169" class="MJXp-mo">[</span><span id="MJXp-Span-170" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-171" class="MJXp-mo">,</span><span id="MJXp-Span-172" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-173" class="MJXp-mo">,</span><span id="MJXp-Span-174" class="MJXp-mi MJXp-italic">k</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-175" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-176" class="MJXp-mo">[</span><span id="MJXp-Span-177" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-178" class="MJXp-mi MJXp-italic">d</span><span style="vertical-align: -0.4em;" id="MJXp-Span-179" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-180" class="MJXp-mo">,</span><span id="MJXp-Span-181" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-182" class="MJXp-mi MJXp-italic">α</span><span style="vertical-align: -0.4em;" id="MJXp-Span-183" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-184" class="MJXp-mo">,</span><span id="MJXp-Span-185" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-186" class="MJXp-mi MJXp-italic">β</span><span style="vertical-align: -0.4em;" id="MJXp-Span-187" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span id="MJXp-Span-188" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-189" class="MJXp-mo">]</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-194" class="MJXp-mi MJXp-italic">m</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-190" class="MJXp-mrow"><span id="MJXp-Span-191" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-192" class="MJXp-mo">=</span><span id="MJXp-Span-193" class="MJXp-mn">1</span></span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-195" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-196" class="MJXp-mo">[</span><span id="MJXp-Span-197" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-198" class="MJXp-mi MJXp-italic">γ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-199" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-200" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-201" class="MJXp-mo">,</span><span id="MJXp-Span-202" class="MJXp-mi MJXp-italic">j</span></span></span><span id="MJXp-Span-203" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-204" class="MJXp-mo">]</span><span style="vertical-align: -0.4em;" id="MJXp-Span-205" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-206" class="MJXp-mn">1</span><span id="MJXp-Span-207" class="MJXp-mo">≤</span><span id="MJXp-Span-208" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-209" class="MJXp-mo"><</span><span id="MJXp-Span-210" class="MJXp-mi MJXp-italic">j</span><span id="MJXp-Span-211" class="MJXp-mo">≤</span><span id="MJXp-Span-212" class="MJXp-mi MJXp-italic">n</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-213" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-214" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-51-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-222" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-223" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-224" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-225" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-226" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-227" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-228" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-229" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-230" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-231" class="mo">[</span><span id="MathJax-Span-232" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.53em, 4.148em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-233" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-234" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-235" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-236" class="msubsup"><span style="display: inline-block; position: relative; width: 0.824em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.55em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-237" class="mi">α</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.552em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-238" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-239" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-240" class="msubsup"><span style="display: inline-block; position: relative; width: 0.778em; height: 0px;"><span style="position: absolute; clip: rect(3.168em, 1000.51em, 4.34em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-241" class="mi">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.506em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-242" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-243" class="msubsup"><span style="display: inline-block; position: relative; width: 1.442em; height: 0px;"><span style="position: absolute; clip: rect(3.184em, 1000.24em, 4.291em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-244" class="mo">]</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.59em, 4.141em, -1000em); top: -4.341em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-245" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.11em, 4.142em, -1000em); top: -3.677em; left: 0.333em;"><span id="MathJax-Span-246" class="texatom"><span id="MathJax-Span-247" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-248" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-249" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-250" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-251" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-252" class="mo">[</span><span id="MathJax-Span-253" class="msubsup"><span style="display: inline-block; position: relative; width: 1.056em; height: 0px;"><span style="position: absolute; clip: rect(3.411em, 1000.44em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-254" class="mi">γ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.028em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.41em;"><span id="MathJax-Span-255" class="texatom"><span id="MathJax-Span-256" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-257" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-258" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-259" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-260" class="msubsup"><span style="display: inline-block; position: relative; width: 2.962em; height: 0px;"><span style="position: absolute; clip: rect(3.184em, 1000.24em, 4.291em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-261" class="mo">]</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.333em;"><span id="MathJax-Span-262" class="texatom"><span id="MathJax-Span-263" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-264" class="mn">1</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-265" class="mo">≤</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-266" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-267" class="mo"><</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-268" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-269" class="mo">≤</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-270" class="mi">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-271" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-272" class="mo">,</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-51" type="math/tex; mode=display">
- [n,K,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le n} ],
- </script></p>
-
-<p>where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-215" class="MJXp-math"><span id="MJXp-Span-216" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-52-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-273" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-274" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-275" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-52" type="math/tex">n</script></span> is the matrix size, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-217" class="MJXp-math"><span id="MJXp-Span-218" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-53-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-276" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-277" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-278" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-53" type="math/tex">K</script></span> is the centre of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-219" class="MJXp-math"><span id="MJXp-Span-220" class="MJXp-mi MJXp-italic">A</span></span></span><span style="" tabindex="0" id="MathJax-Element-54-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-279" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-280" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-281" class="mi">A</span></span></span></span></span></nobr></span><script id="MathJax-Element-54" type="math/tex">A</script></span> (a finite field extension of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-221" class="MJXp-math"><span id="MJXp-Span-222" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-55-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-282" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-283" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-284" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-55" type="math/tex">F</script></span>) and the remaining data are integers whose interpretation is explained in <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84A142407B7565E0"><span class="RefLink">9.12</span></a>.</p>
-
-<p>In some cases (for the zero characteristic coefficient field) the size <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-223" class="MJXp-math"><span id="MJXp-Span-224" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-56-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-285" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-286" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-287" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-56" type="math/tex">n</script></span> of the matrix algebras is not a positive integer but a positive rational number. This is a consequence of the fact that the <em>Brauer-Witt Theorem</em> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBY">[Yam74]</a> only ensures that each <em>Wedderburn component</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of a semisimple group algebra is Brauer equivalent (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to a <em>cyclotomic algebra</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>), but not necessarily isomorphic to a full matrix algebra of a cyclotomic algebra. For example, a Wedderburn component <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-225" class="MJXp-math"><span id="MJXp-Span-226" class="MJXp-mi MJXp-italic">D</span></span></span><span style="" tabindex="0" id="MathJax-Element-57-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-288" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-289" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-290" class="mi">D</span></span></span></span></span></nobr></span><script id="MathJax-Element-57" type="math/tex">D</script></span> of a group algebra can be a division algebra but not a cyclotomic algebra. In this case <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-227" class="MJXp-math"><span id="MJXp-Span-228" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-229" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-230" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-231" class="MJXp-mo">(</span><span id="MJXp-Span-232" class="MJXp-mi MJXp-italic">D</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-233" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-58-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-291" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-292" class="mrow"><span id="MathJax-Span-293" class="msubsup"><span style="display: inline-block; position: relative; width: 1.262em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-294" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-295" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-296" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-297" class="mi">D</span><span style="font-family: STIXGeneral;" id="MathJax-Span-298" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-58" type="math/tex">M_n(D)</script></span> is a cyclotomic algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-234" class="MJXp-math"><span id="MJXp-Span-235" class="MJXp-mi MJXp-italic">C</span></span></span><span style="" tabindex="0" id="MathJax-Element-59-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-299" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-300" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-301" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-59" type="math/tex">C</script></span> for some <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-236" class="MJXp-math"><span id="MJXp-Span-237" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-60-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-302" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-303" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-304" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-60" type="math/tex">n</script></span> and therefore <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-238" class="MJXp-math"><span id="MJXp-Span-239" class="MJXp-mi MJXp-italic">D</span></span></span><span style="" tabindex="0" id="MathJax-Element-61-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-305" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-306" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-307" class="mi">D</span></span></span></span></span></nobr></span><script id="MathJax-Element-61" type="math/tex">D</script></span> can be described as <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-240" class="MJXp-math"><span id="MJXp-Span-241" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-242" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-243" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-244" class="MJXp-mn">1</span><span id="MJXp-Span-245" class="MJXp-mrow"><span id="MJXp-Span-246" class="MJXp-mo">/</span></span><span id="MJXp-Span-247" class="MJXp-mi MJXp-italic">n</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-248" class="MJXp-mo">(</span><span id="MJXp-Span-249" class="MJXp-mi MJXp-italic">C</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-250" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-62-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-308" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-309" class="mrow"><span id="MathJax-Span-310" class="msubsup"><span style="display: inline-block; position: relative; width: 1.812em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-311" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span id="MathJax-Span-312" class="texatom"><span id="MathJax-Span-313" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-314" class="mn">1</span><span id="MathJax-Span-315" class="texatom"><span id="MathJax-Span-316" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-317" class="mo">/</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-318" class="mi">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-319" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-320" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-321" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-62" type="math/tex">M_{1/n}(C)</script></span> (see last Example in <code class="func">WedderburnDecomposition</code> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap2_mj.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>)).</p>
-
-<p>The main algorithm of <strong class="pkg">Wedderga</strong> is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del RÃo and Simón <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBORS">[ORS04]</a> for rational group algebras of <em>strongly monomial groups</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84C694978557EFE5"><span class="RefLink">9.16</span></a>).</p>
-
-<p><a id="X87B6505C7C2EE054" name="X87B6505C7C2EE054"></a></p>
-
-<h4>9.4 <span class="Heading">Characters and primitive central idempotents</span></h4>
-
-<p>A <em>primitive central idempotent</em> of a ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-251" class="MJXp-math"><span id="MJXp-Span-252" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-63-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-322" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-323" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-324" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-63" type="math/tex">R</script></span> is a non-zero central idempotent <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-253" class="MJXp-math"><span id="MJXp-Span-254" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-64-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-325" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-326" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-327" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-64" type="math/tex">e</script></span> which cannot be written as the sum of two non-zero central idempotents of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-255" class="MJXp-math"><span id="MJXp-Span-256" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-257" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-65-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-328" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-329" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-330" class="mi">R</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-331" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-65" type="math/tex">Re</script></span>, or equivalently, such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-258" class="MJXp-math"><span id="MJXp-Span-259" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-260" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-66-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-332" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-333" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-334" class="mi">R</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-335" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-66" type="math/tex">Re</script></span> is indecomposable as a direct product of two non-trivial two-sided ideals.</p>
-
-<p>The <em>Wedderburn components</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of a semisimple ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-261" class="MJXp-math"><span id="MJXp-Span-262" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-67-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-336" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-337" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-338" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-67" type="math/tex">R</script></span> are the rings of the form <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-263" class="MJXp-math"><span id="MJXp-Span-264" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-265" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-68-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-339" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-340" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-341" class="mi">R</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-342" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-68" type="math/tex">Re</script></span> for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-266" class="MJXp-math"><span id="MJXp-Span-267" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-69-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-343" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-344" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-345" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-69" type="math/tex">e</script></span> running over the set of primitive central idempotents of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-268" class="MJXp-math"><span id="MJXp-Span-269" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-70-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-346" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-347" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-348" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-70" type="math/tex">R</script></span>.</p>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-270" class="MJXp-math"><span id="MJXp-Span-271" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-272" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-71-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-349" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-350" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-351" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-352" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-71" type="math/tex">FG</script></span> be a <em>semisimple group algebra</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7FDD93FB79ADCC91"><span class="RefLink">9.2</span></a>) and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-273" class="MJXp-math"><span id="MJXp-Span-274" class="MJXp-mi MJXp-italic">χ</span></span></span><span style="" tabindex="0" id="MathJax-Element-72-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-353" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-354" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-355" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-72" type="math/tex">\chi</script></span> an irreducible character of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-275" class="MJXp-math"><span id="MJXp-Span-276" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-73-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-356" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-357" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-358" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-73" type="math/tex">G</script></span> (in an algebraic closure of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-277" class="MJXp-math"><span id="MJXp-Span-278" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-74-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-359" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-360" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-361" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-74" type="math/tex">F</script></span>). Then there is a unique Wedderburn component <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-279" class="MJXp-math"><span id="MJXp-Span-280" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-281" class="MJXp-mo">=</span><span id="MJXp-Span-282" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-283" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-284" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-285" class="MJXp-mo">(</span><span id="MJXp-Span-286" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-287" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-75-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-362" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-363" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-364" class="mi">A</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-365" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-366" class="msubsup"><span style="display: inline-block; position: relative; width: 1.142em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-367" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-368" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-369" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-370" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-371" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-75" type="math/tex">A=A_F(\chi)</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-288" class="MJXp-math"><span id="MJXp-Span-289" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-290" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-76-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-372" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-373" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-374" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-375" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-76" type="math/tex">FG</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-291" class="MJXp-math"><span id="MJXp-Span-292" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-293" class="MJXp-mo">(</span><span id="MJXp-Span-294" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-295" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-296" class="MJXp-mo">≠</span><span id="MJXp-Span-297" class="MJXp-mn">0</span></span></span><span style="" tabindex="0" id="MathJax-Element-77-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-376" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-377" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-378" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-379" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-380" class="mi">A</span><span style="font-family: STIXGeneral;" id="MathJax-Span-381" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-382" class="mo">≠</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-383" class="mn">0</span></span></span></span></span></nobr></span><script id="MathJax-Element-77" type="math/tex">\chi(A)\ne 0</script></span>. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-298" class="MJXp-math"><span id="MJXp-Span-299" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-300" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-301" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-302" class="MJXp-mo">(</span><span id="MJXp-Span-303" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-304" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-78-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-384" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-385" class="mrow"><span id="MathJax-Span-386" class="msubsup"><span style="display: inline-block; position: relative; width: 0.975em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-387" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-388" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-389" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-390" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-391" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-78" type="math/tex">e_F(\chi)</script></span> denote the unique primitive central idempotent of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-305" class="MJXp-math"><span id="MJXp-Span-306" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-307" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-79-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-392" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-393" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-394" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-395" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-79" type="math/tex">FG</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-308" class="MJXp-math"><span id="MJXp-Span-309" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-310" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-311" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-312" class="MJXp-mo">(</span><span id="MJXp-Span-313" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-314" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-80-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-396" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-397" class="mrow"><span id="MathJax-Span-398" class="msubsup"><span style="display: inline-block; position: relative; width: 1.142em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-399" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-400" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-401" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-402" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-403" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-80" type="math/tex">A_F(\chi)</script></span>, that is the identity of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-315" class="MJXp-math"><span id="MJXp-Span-316" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-317" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-318" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-319" class="MJXp-mo">(</span><span id="MJXp-Span-320" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-321" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-81-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-404" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-405" class="mrow"><span id="MathJax-Span-406" class="msubsup"><span style="display: inline-block; position: relative; width: 1.142em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-407" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-408" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-409" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-410" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-411" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-81" type="math/tex">A_F(\chi)</script></span>, i.e.</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-322" class="MJXp-math MJXp-display"><span id="MJXp-Span-323" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-324" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-325" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-326" class="MJXp-mo">(</span><span id="MJXp-Span-327" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-328" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-329" class="MJXp-mo">=</span><span id="MJXp-Span-330" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-331" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-332" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-333" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-334" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-335" class="MJXp-mo">(</span><span id="MJXp-Span-336" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-337" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-338" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-82-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-412" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-413" class="mrow"><span id="MathJax-Span-414" class="msubsup"><span style="display: inline-block; position: relative; width: 1.142em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-415" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-416" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-417" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-418" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-419" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-420" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-421" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-422" class="mi">G</span><span id="MathJax-Span-423" class="msubsup"><span style="display: inline-block; position: relative; width: 0.975em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-424" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-425" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-426" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-427" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-428" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-429" class="mo">.</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-82" type="math/tex; mode=display">
- A_F(\chi)=FGe_F(\chi).
- </script></p>
-
-<p>The centre of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-339" class="MJXp-math"><span id="MJXp-Span-340" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-341" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-342" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-343" class="MJXp-mo">(</span><span id="MJXp-Span-344" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-345" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-83-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-430" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-431" class="mrow"><span id="MathJax-Span-432" class="msubsup"><span style="display: inline-block; position: relative; width: 1.142em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-433" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-434" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-435" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-436" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-437" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-83" type="math/tex">A_F(\chi)</script></span> is <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-346" class="MJXp-math"><span id="MJXp-Span-347" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-348" class="MJXp-mo">(</span><span id="MJXp-Span-349" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-350" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-351" class="MJXp-mo">=</span><span id="MJXp-Span-352" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-353" class="MJXp-mo">(</span><span id="MJXp-Span-354" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-355" class="MJXp-mo">(</span><span id="MJXp-Span-356" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-357" class="MJXp-mo">)</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-358" class="MJXp-mo">:</span><span id="MJXp-Span-359" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-360" class="MJXp-mo">∈</span><span id="MJXp-Span-361" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-362" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-84-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-438" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-439" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-440" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-441" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-442" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-443" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-444" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-445" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-446" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-447" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-448" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-449" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-450" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-451" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-452" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-453" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-454" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-455" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-84" type="math/tex">F(\chi)=F(\chi(g):g \in G)</script></span>, the <em>field of character values</em> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-363" class="MJXp-math"><span id="MJXp-Span-364" class="MJXp-mi MJXp-italic">χ</span></span></span><span style="" tabindex="0" id="MathJax-Element-85-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-456" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-457" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-458" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-85" type="math/tex">\chi</script></span> over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-365" class="MJXp-math"><span id="MJXp-Span-366" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-86-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-459" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-460" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-461" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-86" type="math/tex">F</script></span>.</p>
-
-<p>The map <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-367" class="MJXp-math"><span id="MJXp-Span-368" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-369" class="MJXp-mo">↦</span><span id="MJXp-Span-370" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-371" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-372" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-373" class="MJXp-mo">(</span><span id="MJXp-Span-374" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-375" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-87-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-462" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-463" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-464" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-465" class="mo">↦</span><span style="padding-left: 0.313em;" id="MathJax-Span-466" class="msubsup"><span style="display: inline-block; position: relative; width: 1.142em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-467" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-468" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-469" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-470" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-471" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-87" type="math/tex">\chi \mapsto A_F(\chi)</script></span> defines a surjective map from the set of irreducible characters of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-376" class="MJXp-math"><span id="MJXp-Span-377" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-88-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-472" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-473" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-474" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-88" type="math/tex">G</script></span> (in an algebraic closure of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-378" class="MJXp-math"><span id="MJXp-Span-379" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-89-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-475" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-476" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-477" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-89" type="math/tex">F</script></span>) onto the set of Wedderburn components of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-380" class="MJXp-math"><span id="MJXp-Span-381" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-382" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-90-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-478" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-479" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-480" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-481" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-90" type="math/tex">FG</script></span>.</p>
-
-<p>Equivalently, the map <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-383" class="MJXp-math"><span id="MJXp-Span-384" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-385" class="MJXp-mo">↦</span><span id="MJXp-Span-386" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-387" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-388" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-389" class="MJXp-mo">(</span><span id="MJXp-Span-390" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-391" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-91-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-482" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-483" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-484" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-485" class="mo">↦</span><span style="padding-left: 0.313em;" id="MathJax-Span-486" class="msubsup"><span style="display: inline-block; position: relative; width: 0.975em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-487" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-488" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-489" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-490" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-491" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-91" type="math/tex">\chi \mapsto e_F(\chi)</script></span> defines a surjective map from the set of irreducible characters of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-392" class="MJXp-math"><span id="MJXp-Span-393" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-92-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-492" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-493" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-494" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-92" type="math/tex">G</script></span> (in an algebraic closure of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-394" class="MJXp-math"><span id="MJXp-Span-395" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-93-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-495" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-496" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-497" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-93" type="math/tex">F</script></span>) onto the set of primitive central idempontents of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-396" class="MJXp-math"><span id="MJXp-Span-397" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-398" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-94-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-498" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-499" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-500" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-501" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-94" type="math/tex">FG</script></span>.</p>
-
-<p>If the irreducible character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-399" class="MJXp-math"><span id="MJXp-Span-400" class="MJXp-mi MJXp-italic">χ</span></span></span><span style="" tabindex="0" id="MathJax-Element-95-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-502" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-503" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-504" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-95" type="math/tex">\chi</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-401" class="MJXp-math"><span id="MJXp-Span-402" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-96-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-505" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-506" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-507" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-96" type="math/tex">G</script></span> takes values in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-403" class="MJXp-math"><span id="MJXp-Span-404" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-97-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-508" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-509" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-510" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-97" type="math/tex">F</script></span> then</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-405" class="MJXp-math MJXp-display"><span id="MJXp-Span-406" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-407" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-408" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-409" class="MJXp-mo">(</span><span id="MJXp-Span-410" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-411" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-412" class="MJXp-mo">=</span><span id="MJXp-Span-413" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-414" class="MJXp-mo">(</span><span id="MJXp-Span-415" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-416" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-417" class="MJXp-mo">=</span><span style="vertical-align: 0.25em;" id="MJXp-Span-418" class="MJXp-mfrac"><span class="MJXp-box"><span id="MJXp-Span-419" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-420" class="MJXp-mo">(</span><span id="MJXp-Span-421" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-422" class="MJXp-mo">)</span></span><span style="margin-top: -0.9em;" class="MJXp-box"><span class="MJXp-denom"><span><span style="height: 1em; border-top: medium none; border-bottom: 1px solid; margin: 0.1em 0px;" class="MJXp-rule"></span></span><span><span class="MJXp-box"><span id="MJXp-Span-423" class="MJXp-mrow"><span style="margin-left: 0.167em; margin-right: 0.167em;" id="MJXp-Span-424" class="MJXp-mo">|</span></span><span id="MJXp-Span-425" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-426" class="MJXp-mrow"><span style="margin-left: 0.167em; margin-right: 0.167em;" id="MJXp-Span-427" class="MJXp-mo">|</span></span></span></span></span></span></span><span id="MJXp-Span-428" class="MJXp-munderover"><span class=""><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-429" class="MJXp-mo"><span class="MJXp-largeop">∑</span></span></span><span class=" MJXp-script"><span style="margin-left: 0px;" id="MJXp-Span-430" class="MJXp-mrow"><span id="MJXp-Span-431" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-432" class="MJXp-mo">∈</span><span id="MJXp-Span-433" class="MJXp-mi MJXp-italic">G</span></span></span></span><span id="MJXp-Span-434" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-435" class="MJXp-mo">(</span><span id="MJXp-Span-436" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-437" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: 0.5em;" id="MJXp-Span-438" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-439" class="MJXp-mo">−</span><span id="MJXp-Span-440" class="MJXp-mn">1</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-441" class="MJXp-mo">)</span><span id="MJXp-Span-442" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-443" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-98-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-511" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-512" class="mrow"><span id="MathJax-Span-513" class="msubsup"><span style="display: inline-block; position: relative; width: 0.975em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-514" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-515" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-516" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-517" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-518" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-519" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-520" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-521" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-522" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-523" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-524" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-525" class="mfrac"><span style="display: inline-block; position: relative; width: 1.784em; height: 0px; margin-right: 0.12em; margin-left: 0.12em;"><span style="position: absolute; clip: rect(3.17em, 1001.62em, 4.342em, -1000em); top: -4.667em; left: 50%; margin-left: -0.832em;"><span id="MathJax-Span-526" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-527" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-528" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-529" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-530" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.17em, 1001.05em, 4.153em, -1000em); top: -3.304em; left: 50%; margin-left: -0.561em;"><span id="MathJax-Span-531" class="mrow"><span id="MathJax-Span-532" class="texatom"><span id="MathJax-Span-533" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-534" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-535" class="mi">G</span><span id="MathJax-Span-536" class="texatom"><span id="MathJax-Span-537" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-538" class="mo">|</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(0.853em, 1001.78em, 1.202em, -1000em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.784em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.058em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-539" class="munderover"><span style="display: inline-block; position: relative; width: 1.348em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0.028em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-540" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.375em, 1001.35em, 4.38em, -1000em); top: -2.84em; left: 0em;"><span id="MathJax-Span-541" class="texatom"><span id="MathJax-Span-542" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-543" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-544" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-545" class="mi">G</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-546" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-547" class="mo">(</span><span id="MathJax-Span-548" class="msubsup"><span style="display: inline-block; position: relative; width: 1.413em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-549" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.5em;"><span id="MathJax-Span-550" class="texatom"><span id="MathJax-Span-551" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-552" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-553" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-554" class="mo">)</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-555" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-556" class="mo">.</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-98" type="math/tex; mode=display">
- e_F(\chi) = e(\chi) = \frac{\chi(1)}{|G|} \sum_{g\in G} \chi(g^{-1}) g.
- </script></p>
-
-<p>In general one has</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-444" class="MJXp-math MJXp-display"><span id="MJXp-Span-445" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-446" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-447" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-448" class="MJXp-mo">(</span><span id="MJXp-Span-449" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-450" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-451" class="MJXp-mo">=</span><span id="MJXp-Span-452" class="MJXp-munderover"><span class=""><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-453" class="MJXp-mo"><span class="MJXp-largeop">∑</span></span></span><span class=" MJXp-script"><span style="margin-left: 0px;" id="MJXp-Span-454" class="MJXp-mrow"><span id="MJXp-Span-455" class="MJXp-mi MJXp-italic">σ</span><span id="MJXp-Span-456" class="MJXp-mo">∈</span><span id="MJXp-Span-457" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-458" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-459" class="MJXp-mi MJXp-italic">l</span><span id="MJXp-Span-460" class="MJXp-mo">(</span><span id="MJXp-Span-461" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-462" class="MJXp-mo">(</span><span id="MJXp-Span-463" class="MJXp-mi MJXp-italic">χ</span><span id="MJXp-Span-464" class="MJXp-mo">)</span><span id="MJXp-Span-465" class="MJXp-mrow"><span id="MJXp-Span-466" class="MJXp-mo">/</span></span><span id="MJXp-Span-467" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-468" class="MJXp-mo">)</span></span></span></span><span id="MJXp-Span-469" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-470" class="MJXp-mo">(</span><span id="MJXp-Span-471" class="MJXp-mi MJXp-italic">σ</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-472" class="MJXp-mo">∘</span><span id="MJXp-Span-473" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-474" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-475" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-99-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-557" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-558" class="mrow"><span id="MathJax-Span-559" class="msubsup"><span style="display: inline-block; position: relative; width: 0.975em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-560" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-561" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-562" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-563" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-564" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-565" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-566" class="munderover"><span style="display: inline-block; position: relative; width: 4.325em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 1.516em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-567" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1004.29em, 4.381em, -1000em); top: -2.84em; left: 0em;"><span id="MathJax-Span-568" class="texatom"><span id="MathJax-Span-569" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-570" class="mi">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-571" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-572" class="mi">G</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-573" class="mi">a</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-574" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-575" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-576" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-577" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-578" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.029em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-579" class="mo">)</span><span id="MathJax-Span-580" class="texatom"><span id="MathJax-Span-581" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-582" class="mo">/</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-583" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-584" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-585" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-586" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-587" class="mi">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.033em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-588" class="mo">∘</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-589" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-590" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-591" class="mo">.</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-99" type="math/tex; mode=display">
- e_F(\chi) = \sum_{\sigma \in Gal(F(\chi)/F)} e(\sigma \circ \chi).
- </script></p>
-
-<p><a id="X7A24D5407F72C633" name="X7A24D5407F72C633"></a></p>
-
-<h4>9.5 <span class="Heading">Central simple algebras and Brauer equivalence</span></h4>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-476" class="MJXp-math"><span id="MJXp-Span-477" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-100-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-592" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-593" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-594" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-100" type="math/tex">K</script></span> be a field. A <em>central simple <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-478" class="MJXp-math"><span id="MJXp-Span-479" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-101-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-595" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-596" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-597" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-101" type="math/tex">K</script></span>-algebra</em> is a finite dimensional <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-480" class="MJXp-math"><span id="MJXp-Span-481" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-102-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-598" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-599" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-600" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-102" type="math/tex">K</script></span>-algebra with center <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-482" class="MJXp-math"><span id="MJXp-Span-483" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-103-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-601" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-602" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-603" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-103" type="math/tex">K</script></span> which has no non-trivial proper ideals. Every central simple <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-484" class="MJXp-math"><span id="MJXp-Span-485" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-104-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-604" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-605" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-606" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-104" type="math/tex">K</script></span>-algebra is isomorphic to a matrix algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-486" class="MJXp-math"><span id="MJXp-Span-487" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-488" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-489" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-490" class="MJXp-mo">(</span><span id="MJXp-Span-491" class="MJXp-mi MJXp-italic">D</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-492" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-105-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-607" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-608" class="mrow"><span id="MathJax-Span-609" class="msubsup"><span style="display: inline-block; position: relative; width: 1.262em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-610" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-611" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-612" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-613" class="mi">D</span><span style="font-family: STIXGeneral;" id="MathJax-Span-614" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-105" type="math/tex">M_n(D)</script></span> where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-493" class="MJXp-math"><span id="MJXp-Span-494" class="MJXp-mi MJXp-italic">D</span></span></span><span style="" tabindex="0" id="MathJax-Element-106-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-615" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-616" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-617" class="mi">D</span></span></span></span></span></nobr></span><script id="MathJax-Element-106" type="math/tex">D</script></span> is a division algebra (which is finite-dimensional over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-495" class="MJXp-math"><span id="MJXp-Span-496" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-107-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-618" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-619" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-620" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-107" type="math/tex">K</script></span> and has centre <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-497" class="MJXp-math"><span id="MJXp-Span-498" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-108-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-621" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-622" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-623" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-108" type="math/tex">K</script></span>). The division algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-499" class="MJXp-math"><span id="MJXp-Span-500" class="MJXp-mi MJXp-italic">D</span></span></span><span style="" tabindex="0" id="MathJax-Element-109-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-624" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-625" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-626" class="mi">D</span></span></span></span></span></nobr></span><script id="MathJax-Element-109" type="math/tex">D</script></span> is unique up to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-501" class="MJXp-math"><span id="MJXp-Span-502" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-110-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-627" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-628" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-629" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-110" type="math/tex">K</script></span>-isomorphisms.</p>
-
-<p>Two central simple <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-503" class="MJXp-math"><span id="MJXp-Span-504" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-111-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-630" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-631" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-632" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-111" type="math/tex">K</script></span>-algebras <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-505" class="MJXp-math"><span id="MJXp-Span-506" class="MJXp-mi MJXp-italic">A</span></span></span><span style="" tabindex="0" id="MathJax-Element-112-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-633" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-634" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-635" class="mi">A</span></span></span></span></span></nobr></span><script id="MathJax-Element-112" type="math/tex">A</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-507" class="MJXp-math"><span id="MJXp-Span-508" class="MJXp-mi MJXp-italic">B</span></span></span><span style="" tabindex="0" id="MathJax-Element-113-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-636" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-637" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-638" class="mi">B</span></span></span></span></span></nobr></span><script id="MathJax-Element-113" type="math/tex">B</script></span> are said to be <em>Brauer equivalent</em>, or simply <em>equivalent</em>, if there is a division algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-509" class="MJXp-math"><span id="MJXp-Span-510" class="MJXp-mi MJXp-italic">D</span></span></span><span style="" tabindex="0" id="MathJax-Element-114-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-639" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-640" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-641" class="mi">D</span></span></span></span></span></nobr></span><script id="MathJax-Element-114" type="math/tex">D</script></span> and two positive integers <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-511" class="MJXp-math"><span id="MJXp-Span-512" class="MJXp-mi MJXp-italic">m</span></span></span><span style="" tabindex="0" id="MathJax-Element-115-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-642" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-643" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-644" class="mi">m</span></span></span></span></span></nobr></span><script id="MathJax-Element-115" type="math/tex">m</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-513" class="MJXp-math"><span id="MJXp-Span-514" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-116-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-645" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-646" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-647" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-116" type="math/tex">n</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-515" class="MJXp-math"><span id="MJXp-Span-516" class="MJXp-mi MJXp-italic">A</span></span></span><span style="" tabindex="0" id="MathJax-Element-117-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-648" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-649" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-650" class="mi">A</span></span></span></span></span></nobr></span><script id="MathJax-Element-117" type="math/tex">A</script></span> is isomorphic to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-517" class="MJXp-math"><span id="MJXp-Span-518" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-519" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-520" class="MJXp-mi MJXp-italic MJXp-script">m</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-521" class="MJXp-mo">(</span><span id="MJXp-Span-522" class="MJXp-mi MJXp-italic">D</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-523" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-118-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-651" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-652" class="mrow"><span id="MathJax-Span-653" class="msubsup"><span style="display: inline-block; position: relative; width: 1.419em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-654" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-655" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-656" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-657" class="mi">D</span><span style="font-family: STIXGeneral;" id="MathJax-Span-658" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-118" type="math/tex">M_m(D)</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-524" class="MJXp-math"><span id="MJXp-Span-525" class="MJXp-mi MJXp-italic">B</span></span></span><span style="" tabindex="0" id="MathJax-Element-119-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-659" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-660" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-661" class="mi">B</span></span></span></span></span></nobr></span><script id="MathJax-Element-119" type="math/tex">B</script></span> is isomorphic to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-526" class="MJXp-math"><span id="MJXp-Span-527" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-528" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-529" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-530" class="MJXp-mo">(</span><span id="MJXp-Span-531" class="MJXp-mi MJXp-italic">D</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-532" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-120-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-662" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-663" class="mrow"><span id="MathJax-Span-664" class="msubsup"><span style="display: inline-block; position: relative; width: 1.262em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-665" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-666" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-667" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-668" class="mi">D</span><span style="font-family: STIXGeneral;" id="MathJax-Span-669" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-120" type="math/tex">M_n(D)</script></span>.</p>
-
-<p><a id="X7FB21779832CE1CB" name="X7FB21779832CE1CB"></a></p>
-
-<h4>9.6 <span class="Heading">Crossed Products</span></h4>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-533" class="MJXp-math"><span id="MJXp-Span-534" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-121-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-670" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-671" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-672" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-121" type="math/tex">R</script></span> be a ring and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-535" class="MJXp-math"><span id="MJXp-Span-536" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-122-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-673" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-674" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-675" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-122" type="math/tex">G</script></span> a group.</p>
-
-<p><strong class="button">Intrinsic definition</strong>. A <em>crossed product</em> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBP">[Pas89]</a> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-537" class="MJXp-math"><span id="MJXp-Span-538" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-123-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-676" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-677" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-678" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-123" type="math/tex">G</script></span> over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-539" class="MJXp-math"><span id="MJXp-Span-540" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-124-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-679" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-680" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-681" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-124" type="math/tex">R</script></span> (or with coefficients in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-541" class="MJXp-math"><span id="MJXp-Span-542" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-125-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-682" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-683" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-684" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-125" type="math/tex">R</script></span>) is a ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-543" class="MJXp-math"><span id="MJXp-Span-544" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-545" class="MJXp-mo">∗</span><span id="MJXp-Span-546" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-126-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-685" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-686" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-687" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-688" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-689" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-126" type="math/tex">R*G</script></span> with a decomposition into a direct sum of additive subgroups</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-547" class="MJXp-math MJXp-display"><span id="MJXp-Span-548" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-549" class="MJXp-mo">∗</span><span id="MJXp-Span-550" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-551" class="MJXp-mo">=</span><span id="MJXp-Span-552" class="MJXp-munderover"><span class=""><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-553" class="MJXp-mo"><span class="MJXp-largeop">â¨</span></span></span><span class=" MJXp-script"><span style="margin-left: 0px;" id="MJXp-Span-554" class="MJXp-mrow"><span id="MJXp-Span-555" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-556" class="MJXp-mo">∈</span><span id="MJXp-Span-557" class="MJXp-mi MJXp-italic">G</span></span></span></span><span id="MJXp-Span-558" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-559" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-560" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-127-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-690" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-691" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-692" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-693" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-694" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-695" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-696" class="munderover"><span style="display: inline-block; position: relative; width: 1.555em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.5em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-697" class="mo">â¨</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.375em, 1001.35em, 4.38em, -1000em); top: -2.84em; left: 0.103em;"><span id="MathJax-Span-698" class="texatom"><span id="MathJax-Span-699" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-700" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-701" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-702" class="mi">G</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-703" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-704" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-705" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span></div><script id="MathJax-Element-127" type="math/tex; mode=display">
- R*G = \bigoplus_{g \in G} A_g
- </script></p>
-
-<p>such that for each <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-561" class="MJXp-math"><span id="MJXp-Span-562" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-563" class="MJXp-mo">,</span><span id="MJXp-Span-564" class="MJXp-mi MJXp-italic">h</span></span></span><span style="" tabindex="0" id="MathJax-Element-128-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-706" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-707" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-708" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-709" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-710" class="mi">h</span></span></span></span></span></nobr></span><script id="MathJax-Element-128" type="math/tex">g,h</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-565" class="MJXp-math"><span id="MJXp-Span-566" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-129-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-711" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-712" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-713" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-129" type="math/tex">G</script></span> one has:</p>
-
-<p>* <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-567" class="MJXp-math"><span id="MJXp-Span-568" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-569" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-570" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-571" class="MJXp-mo">=</span><span id="MJXp-Span-572" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-130-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-714" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-715" class="mrow"><span id="MathJax-Span-716" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-717" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-718" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-719" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-720" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-130" type="math/tex">A_1=R</script></span> (here <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-573" class="MJXp-math"><span id="MJXp-Span-574" class="MJXp-mn">1</span></span></span><span style="" tabindex="0" id="MathJax-Element-131-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-721" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-722" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-723" class="mn">1</span></span></span></span></span></nobr></span><script id="MathJax-Element-131" type="math/tex">1</script></span> denotes the identity of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-575" class="MJXp-math"><span id="MJXp-Span-576" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-132-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-724" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-725" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-726" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-132" type="math/tex">G</script></span>),</p>
-
-<p>* <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-577" class="MJXp-math"><span id="MJXp-Span-578" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-579" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-580" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span id="MJXp-Span-581" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-582" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-583" class="MJXp-mi MJXp-italic MJXp-script">h</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-584" class="MJXp-mo">=</span><span id="MJXp-Span-585" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-586" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-587" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-588" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-589" class="MJXp-mi MJXp-italic">h</span></span></span></span></span><span style="" tabindex="0" id="MathJax-Element-133-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-727" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-728" class="mrow"><span id="MathJax-Span-729" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-730" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-731" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-732" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-733" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-734" class="mi">h</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-735" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-736" class="msubsup"><span style="display: inline-block; position: relative; width: 1.393em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-737" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span id="MathJax-Span-738" class="texatom"><span id="MathJax-Span-739" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-740" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-741" class="mi">h</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-133" type="math/tex">A_g A_h = A_{gh}</script></span> and</p>
-
-<p>* <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-590" class="MJXp-math"><span id="MJXp-Span-591" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-592" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-593" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-134-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-742" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-743" class="mrow"><span id="MathJax-Span-744" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-745" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-746" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-134" type="math/tex">A_g</script></span> has a unit of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-594" class="MJXp-math"><span id="MJXp-Span-595" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-596" class="MJXp-mo">∗</span><span id="MJXp-Span-597" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-135-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-747" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-748" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-749" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-750" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-751" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-135" type="math/tex">R*G</script></span>.</p>
-
-<p><strong class="button">Extrinsic definition</strong>. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-598" class="MJXp-math"><span id="MJXp-Span-599" class="MJXp-mi MJXp-italic">A</span><span id="MJXp-Span-600" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-601" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-602" class="MJXp-mo">(</span><span id="MJXp-Span-603" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-604" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-136-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-752" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-753" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-754" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-755" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-756" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-757" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-758" class="mi">R</span><span style="font-family: STIXGeneral;" id="MathJax-Span-759" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-136" type="math/tex">Aut(R)</script></span> denote the group of automorphisms of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-605" class="MJXp-math"><span id="MJXp-Span-606" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-137-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-760" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-761" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-762" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-137" type="math/tex">R</script></span> and let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-607" class="MJXp-math"><span id="MJXp-Span-608" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-609" class="MJXp-mi MJXp-italic">R</span><span style="vertical-align: 0.5em;" id="MJXp-Span-610" class="MJXp-mo MJXp-script">∗</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-138-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-763" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-764" class="mrow"><span id="MathJax-Span-765" class="msubsup"><span style="display: inline-block; position: relative; width: 1.056em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.59em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-766" class="mi">R</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-767" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-138" type="math/tex">R^*</script></span> denote the group of units of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-611" class="MJXp-math"><span id="MJXp-Span-612" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-139-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-768" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-769" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-770" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-139" type="math/tex">R</script></span>.</p>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-613" class="MJXp-math"><span id="MJXp-Span-614" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-615" class="MJXp-mo">:</span><span id="MJXp-Span-616" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-617" class="MJXp-mo">→</span><span id="MJXp-Span-618" class="MJXp-mi MJXp-italic">A</span><span id="MJXp-Span-619" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-620" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-621" class="MJXp-mo">(</span><span id="MJXp-Span-622" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-623" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-140-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-771" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-772" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-773" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-774" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-775" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-776" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-777" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-778" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-779" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-780" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-781" class="mi">R</span><span style="font-family: STIXGeneral;" id="MathJax-Span-782" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-140" type="math/tex">a:G \rightarrow Aut(R)</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-624" class="MJXp-math"><span id="MJXp-Span-625" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-626" class="MJXp-mo">:</span><span id="MJXp-Span-627" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-628" class="MJXp-mo">×</span><span id="MJXp-Span-629" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-630" class="MJXp-mo">→</span><span id="MJXp-Span-631" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-632" class="MJXp-mi MJXp-italic">R</span><span style="vertical-align: 0.5em;" id="MJXp-Span-633" class="MJXp-mo MJXp-script">∗</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-141-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-783" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-784" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-785" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-786" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-787" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-788" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-789" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-790" class="mo">→</span><span style="padding-left: 0.313em;" id="MathJax-Span-791" class="msubsup"><span style="display: inline-block; position: relative; width: 1.056em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.59em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-792" class="mi">R</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-793" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-141" type="math/tex">t:G \times G \rightarrow R^*</script></span> be mappings satisfying the following conditions for every <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-634" class="MJXp-math"><span id="MJXp-Span-635" class="MJXp-mi MJXp-italic">g</span></span></span><span style="" tabindex="0" id="MathJax-Element-142-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-794" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-795" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-796" class="mi">g</span></span></span></span></span></nobr></span><script id="MathJax-Element-142" type="math/tex">g</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-636" class="MJXp-math"><span id="MJXp-Span-637" class="MJXp-mi MJXp-italic">h</span></span></span><span style="" tabindex="0" id="MathJax-Element-143-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-797" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-798" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-799" class="mi">h</span></span></span></span></span></nobr></span><script id="MathJax-Element-143" type="math/tex">h</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-638" class="MJXp-math"><span id="MJXp-Span-639" class="MJXp-mi MJXp-italic">k</span></span></span><span style="" tabindex="0" id="MathJax-Element-144-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-800" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-801" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-802" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-144" type="math/tex">k</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-640" class="MJXp-math"><span id="MJXp-Span-641" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-145-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-803" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-804" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-805" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-145" type="math/tex">G</script></span>:</p>
-
-<p>(1) <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-642" class="MJXp-math"><span id="MJXp-Span-643" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-644" class="MJXp-mo">(</span><span id="MJXp-Span-645" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-646" class="MJXp-mi MJXp-italic">h</span><span id="MJXp-Span-647" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-648" class="MJXp-mo">)</span><span style="vertical-align: 0.5em;" id="MJXp-Span-649" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-650" class="MJXp-mo">−</span><span id="MJXp-Span-651" class="MJXp-mn">1</span></span></span><span id="MJXp-Span-652" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-653" class="MJXp-mo">(</span><span id="MJXp-Span-654" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-655" class="MJXp-mo">)</span><span id="MJXp-Span-656" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-657" class="MJXp-mo">(</span><span id="MJXp-Span-658" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-659" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-146-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-806" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-807" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-808" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-809" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-810" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-811" class="mi">h</span><span id="MathJax-Span-812" class="msubsup"><span style="display: inline-block; position: relative; width: 1.246em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-813" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span id="MathJax-Span-814" class="texatom"><span id="MathJax-Span-815" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-816" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-817" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-818" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-819" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-820" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-821" class="mo">)</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-822" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-823" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-824" class="mi">h</span><span style="font-family: STIXGeneral;" id="MathJax-Span-825" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-146" type="math/tex">a(gh)^{-1} a(g) a(h)</script></span> is the inner automorphism of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-660" class="MJXp-math"><span id="MJXp-Span-661" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-147-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-826" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-827" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-828" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-147" type="math/tex">R</script></span> induced by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-662" class="MJXp-math"><span id="MJXp-Span-663" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-664" class="MJXp-mo">(</span><span id="MJXp-Span-665" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-666" class="MJXp-mo">,</span><span id="MJXp-Span-667" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-668" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-148-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-829" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-830" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-831" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-832" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-833" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-834" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-835" class="mi">h</span><span style="font-family: STIXGeneral;" id="MathJax-Span-836" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-148" type="math/tex">t(g,h)</script></span> (i.e. the automorphism <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-669" class="MJXp-math"><span id="MJXp-Span-670" class="MJXp-mi MJXp-italic">x</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-671" class="MJXp-mo">↦</span><span id="MJXp-Span-672" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-673" class="MJXp-mo">(</span><span id="MJXp-Span-674" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-675" class="MJXp-mo">,</span><span id="MJXp-Span-676" class="MJXp-mi MJXp-italic">h</span><span id="MJXp-Span-677" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-678" class="MJXp-mo">)</span><span style="vertical-align: 0.5em;" id="MJXp-Span-679" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-680" class="MJXp-mo">−</span><span id="MJXp-Span-681" class="MJXp-mn">1</span></span></span><span id="MJXp-Span-682" class="MJXp-mi MJXp-italic">x</span><span id="MJXp-Span-683" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-684" class="MJXp-mo">(</span><span id="MJXp-Span-685" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-686" class="MJXp-mo">,</span><span id="MJXp-Span-687" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-688" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-149-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-837" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-838" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-839" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-840" class="mo">↦</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-841" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-842" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-843" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-844" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-845" class="mi">h</span><span id="MathJax-Span-846" class="msubsup"><span style="display: inline-block; position: relative; width: 1.246em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-847" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span id="MathJax-Span-848" class="texatom"><span id="MathJax-Span-849" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-850" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-851" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-852" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-853" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-854" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-855" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-856" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-857" class="mi">h</span><span style="font-family: STIXGeneral;" id="MathJax-Span-858" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-149" type="math/tex">x\mapsto t(g,h)^{-1} x t(g,h)</script></span>) and</p>
-
-<p>(2) <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-689" class="MJXp-math"><span id="MJXp-Span-690" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-691" class="MJXp-mo">(</span><span id="MJXp-Span-692" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-693" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-694" class="MJXp-mo">,</span><span id="MJXp-Span-695" class="MJXp-mi MJXp-italic">k</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-696" class="MJXp-mo">)</span><span id="MJXp-Span-697" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-698" class="MJXp-mo">(</span><span id="MJXp-Span-699" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-700" class="MJXp-mo">,</span><span id="MJXp-Span-701" class="MJXp-mi MJXp-italic">h</span><span id="MJXp-Span-702" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-703" class="MJXp-mo">)</span><span style="vertical-align: 0.5em;" id="MJXp-Span-704" class="MJXp-mi MJXp-italic MJXp-script">k</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-705" class="MJXp-mo">=</span><span id="MJXp-Span-706" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-707" class="MJXp-mo">(</span><span id="MJXp-Span-708" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-709" class="MJXp-mo">,</span><span id="MJXp-Span-710" class="MJXp-mi MJXp-italic">h</span><span id="MJXp-Span-711" class="MJXp-mi MJXp-italic">k</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-712" class="MJXp-mo">)</span><span id="MJXp-Span-713" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-714" class="MJXp-mo">(</span><span id="MJXp-Span-715" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-716" class="MJXp-mo">,</span><span id="MJXp-Span-717" class="MJXp-mi MJXp-italic">k</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-718" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-150-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-859" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-860" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-861" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-862" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-863" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-864" class="mi">h</span><span style="font-family: STIXGeneral;" id="MathJax-Span-865" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-866" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-867" class="mo">)</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-868" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-869" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-870" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-871" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-872" class="mi">h</span><span id="MathJax-Span-873" class="msubsup"><span style="display: inline-block; position: relative; width: 0.734em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-874" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-875" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-876" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-877" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-878" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-879" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-880" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-881" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-882" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-883" class="mo">)</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-884" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-885" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-886" class="mi">h</span><span style="font-family: STIXGeneral;" id="MathJax-Span-887" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-888" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-889" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-150" type="math/tex">t(gh,k) t(g,h)^k = t(g,hk) t(h,k)</script></span>, where for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-719" class="MJXp-math"><span id="MJXp-Span-720" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-721" class="MJXp-mo">∈</span><span id="MJXp-Span-722" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-151-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-890" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-891" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-892" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-893" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-894" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-151" type="math/tex">g \in G</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-723" class="MJXp-math"><span id="MJXp-Span-724" class="MJXp-mi MJXp-italic">x</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-725" class="MJXp-mo">∈</span><span id="MJXp-Span-726" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-152-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-895" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-896" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-897" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-898" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-899" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-152" type="math/tex">x \in R</script></span> we denote <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-727" class="MJXp-math"><span id="MJXp-Span-728" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-729" class="MJXp-mo">(</span><span id="MJXp-Span-730" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-731" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-732" class="MJXp-mo">(</span><span id="MJXp-Span-733" class="MJXp-mi MJXp-italic">x</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-734" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-153-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-900" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-901" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-902" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-903" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-904" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-905" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-906" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-907" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-908" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-153" type="math/tex">a(g)(x)</script></span> by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-735" class="MJXp-math"><span id="MJXp-Span-736" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-737" class="MJXp-mi MJXp-italic">x</span><span style="vertical-align: 0.5em;" id="MJXp-Span-738" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-154-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-909" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-910" class="mrow"><span id="MathJax-Span-911" class="msubsup"><span style="display: inline-block; position: relative; width: 0.926em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-912" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.498em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-913" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-154" type="math/tex">x^g</script></span>.</p>
-
-<p>The <em>crossed product</em> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBP">[Pas89]</a> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-739" class="MJXp-math"><span id="MJXp-Span-740" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-155-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-914" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-915" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-916" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-155" type="math/tex">G</script></span> over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-741" class="MJXp-math"><span id="MJXp-Span-742" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-156-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-917" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-918" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-919" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-156" type="math/tex">R</script></span> (or with coefficients in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-743" class="MJXp-math"><span id="MJXp-Span-744" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-157-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-920" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-921" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-922" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-157" type="math/tex">R</script></span>), action <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-745" class="MJXp-math"><span id="MJXp-Span-746" class="MJXp-mi MJXp-italic">a</span></span></span><span style="" tabindex="0" id="MathJax-Element-158-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-923" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-924" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-925" class="mi">a</span></span></span></span></span></nobr></span><script id="MathJax-Element-158" type="math/tex">a</script></span> and twisting <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-747" class="MJXp-math"><span id="MJXp-Span-748" class="MJXp-mi MJXp-italic">t</span></span></span><span style="" tabindex="0" id="MathJax-Element-159-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-926" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-927" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-928" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-159" type="math/tex">t</script></span> is the ring</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-749" class="MJXp-math MJXp-display"><span id="MJXp-Span-750" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-751" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-752" class="MJXp-mo">∗</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-754" class="MJXp-mi MJXp-italic">t</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-753" class="MJXp-mi MJXp-italic">a</span></span></span></span></span></span><span id="MJXp-Span-755" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-756" class="MJXp-mo">=</span><span id="MJXp-Span-757" class="MJXp-munderover"><span class=""><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-758" class="MJXp-mo"><span class="MJXp-largeop">â¨</span></span></span><span class=" MJXp-script"><span style="margin-left: 0px;" id="MJXp-Span-759" class="MJXp-mrow"><span id="MJXp-Span-760" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-761" class="MJXp-mo">∈</span><span id="MJXp-Span-762" class="MJXp-mi MJXp-italic">G</span></span></span></span><span id="MJXp-Span-763" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-764" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-765" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span id="MJXp-Span-766" class="MJXp-mi MJXp-italic">R</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-160-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-929" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-930" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-931" class="mi">R</span><span style="padding-left: 0.25em;" id="MathJax-Span-932" class="msubsup"><span style="display: inline-block; position: relative; width: 0.952em; height: 0px;"><span style="position: absolute; clip: rect(3.375em, 1000.46em, 4.102em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-933" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.46em, 1000.28em, 4.142em, -1000em); top: -4.343em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-934" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.013em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.142em, -1000em); top: -3.843em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-935" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-936" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-937" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-938" class="munderover"><span style="display: inline-block; position: relative; width: 1.555em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.5em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-939" class="mo">â¨</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.375em, 1001.35em, 4.38em, -1000em); top: -2.84em; left: 0.103em;"><span id="MathJax-Span-940" class="texatom"><span id="MathJax-Span-941" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-942" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-943" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-944" class="mi">G</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-945" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-946" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-947" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-948" class="mi">R</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-160" type="math/tex; mode=display">
- R*_a^t G = \bigoplus_{g\in G} u_g R
- </script></p>
-
-<p>where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-767" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-768" class="MJXp-mo">{</span><span id="MJXp-Span-769" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-770" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-771" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-772" class="MJXp-mo">:</span><span id="MJXp-Span-773" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-774" class="MJXp-mo">∈</span><span id="MJXp-Span-775" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-776" class="MJXp-mo">}</span></span></span><span style="" tabindex="0" id="MathJax-Element-161-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-949" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-950" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-951" class="mo">{</span><span id="MathJax-Span-952" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-953" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-954" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-955" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-956" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-957" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-958" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-959" class="mo">}</span></span></span></span></span></nobr></span><script id="MathJax-Element-161" type="math/tex">\{u_g : g\in G \}</script></span> is a set of symbols in one-to-one correspondence with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-777" class="MJXp-math"><span id="MJXp-Span-778" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-162-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-960" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-961" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-962" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-162" type="math/tex">G</script></span>, with addition and multiplication defined by</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-779" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-780" class="MJXp-mo">(</span><span id="MJXp-Span-781" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-782" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-783" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span id="MJXp-Span-784" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-785" class="MJXp-mo">)</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-786" class="MJXp-mo">+</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-787" class="MJXp-mo">(</span><span id="MJXp-Span-788" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-789" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-790" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span id="MJXp-Span-791" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-792" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-793" class="MJXp-mo">=</span><span id="MJXp-Span-794" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-795" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-796" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-797" class="MJXp-mo">(</span><span id="MJXp-Span-798" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-799" class="MJXp-mo">+</span><span id="MJXp-Span-800" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-801" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-802" class="MJXp-mo">,</span><span style="width: 1em; height: 0em;" id="MJXp-Span-803" class="MJXp-mspace"></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-804" class="MJXp-mo">(</span><span id="MJXp-Span-805" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-806" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-807" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span id="MJXp-Span-808" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-809" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-810" class="MJXp-mo">(</span><span id="MJXp-Span-811" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-812" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-813" class="MJXp-mi MJXp-italic MJXp-script">h</span></span><span id="MJXp-Span-814" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-815" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-816" class="MJXp-mo">=</span><span id="MJXp-Span-817" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-818" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-819" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-820" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-821" class="MJXp-mi MJXp-italic">h</span></span></span><span id="MJXp-Span-822" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-823" class="MJXp-mo">(</span><span id="MJXp-Span-824" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-825" class="MJXp-mo">,</span><span id="MJXp-Span-826" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-827" class="MJXp-mo">)</span><span id="MJXp-Span-828" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-829" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: 0.5em;" id="MJXp-Span-830" class="MJXp-mi MJXp-italic MJXp-script">h</span></span><span id="MJXp-Span-831" class="MJXp-mi MJXp-italic">s</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-163-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-963" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-964" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-965" class="mo">(</span><span id="MathJax-Span-966" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-967" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-968" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-969" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-970" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-971" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-972" class="mo">(</span><span id="MathJax-Span-973" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-974" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-975" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-976" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-977" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-978" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-979" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-980" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-981" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-982" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-983" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-984" class="mo">+</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-985" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-986" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-987" class="mo">,</span><span style="height: 0em; vertical-align: 0em; width: 1.125em; display: inline-block; overflow: hidden;" id="MathJax-Span-988" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-989" class="mo">(</span><span id="MathJax-Span-990" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-991" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-992" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-993" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-994" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-995" class="mo">(</span><span id="MathJax-Span-996" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-997" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-998" class="mi">h</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-999" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1000" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1001" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1002" class="msubsup"><span style="display: inline-block; position: relative; width: 1.282em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1003" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-1004" class="texatom"><span id="MathJax-Span-1005" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1006" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1007" class="mi">h</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1008" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1009" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1010" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1011" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1012" class="mi">h</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1013" class="mo">)</span><span id="MathJax-Span-1014" class="msubsup"><span style="display: inline-block; position: relative; width: 0.897em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1015" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.469em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1016" class="mi">h</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1017" class="mi">s</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-163" type="math/tex; mode=display">
- (u_g r) + (u_g s) = u_g(r+s), \quad (u_g r)(u_h s) = u_{gh} t(g,h) r^h s
- </script></p>
-
-<p>for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-832" class="MJXp-math"><span id="MJXp-Span-833" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-834" class="MJXp-mo">,</span><span id="MJXp-Span-835" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-836" class="MJXp-mo">∈</span><span id="MJXp-Span-837" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-164-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1018" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1019" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1020" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1021" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1022" class="mi">h</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1023" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1024" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-164" type="math/tex">g,h \in G</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-838" class="MJXp-math"><span id="MJXp-Span-839" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-840" class="MJXp-mo">,</span><span id="MJXp-Span-841" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-842" class="MJXp-mo">∈</span><span id="MJXp-Span-843" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-165-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1025" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1026" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1027" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1028" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1029" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1030" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1031" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-165" type="math/tex">r,s\in R</script></span>, and extended to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-844" class="MJXp-math"><span id="MJXp-Span-845" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-846" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-847" class="MJXp-mo">∗</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-849" class="MJXp-mi MJXp-italic">t</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-848" class="MJXp-mi MJXp-italic">a</span></span></span></span></span></span><span id="MJXp-Span-850" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-166-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1032" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1033" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1034" class="mi">R</span><span style="padding-left: 0.25em;" id="MathJax-Span-1035" class="msubsup"><span style="display: inline-block; position: relative; width: 0.952em; height: 0px;"><span style="position: absolute; clip: rect(3.375em, 1000.46em, 4.102em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-1036" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.46em, 1000.28em, 4.142em, -1000em); top: -4.343em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1037" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.013em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.142em, -1000em); top: -3.843em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1038" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1039" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-166" type="math/tex">R*_a^t G</script></span> by linearity.</p>
-
-<p>The associativity of the product defined is a consequence of conditions (1) and (2) <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBP">[Pas89]</a>.</p>
-
-<p><strong class="button">Equivalence of the two definitions</strong>. Obviously the crossed product of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-851" class="MJXp-math"><span id="MJXp-Span-852" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-167-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1040" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1041" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1042" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-167" type="math/tex">G</script></span> over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-853" class="MJXp-math"><span id="MJXp-Span-854" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-168-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1043" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1044" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1045" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-168" type="math/tex">R</script></span> defined using the extrinsic definition is a crossed product of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-855" class="MJXp-math"><span id="MJXp-Span-856" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-169-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1046" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1047" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1048" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-169" type="math/tex">G</script></span> over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-857" class="MJXp-math"><span id="MJXp-Span-858" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-859" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-860" class="MJXp-mn MJXp-script">1</span></span><span id="MJXp-Span-861" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-170-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1049" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1050" class="mrow"><span id="MathJax-Span-1051" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1052" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1053" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1054" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-170" type="math/tex">u_1 R</script></span> in the sense of the first definition. Moreover, there is <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-862" class="MJXp-math"><span id="MJXp-Span-863" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-864" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: -0.4em;" id="MJXp-Span-865" class="MJXp-mn MJXp-script">0</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-171-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1055" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1056" class="mrow"><span id="MathJax-Span-1057" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1058" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1059" class="mn">0</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-171" type="math/tex">r_0</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-866" class="MJXp-math"><span id="MJXp-Span-867" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-868" class="MJXp-mi MJXp-italic">R</span><span style="vertical-align: 0.5em;" id="MJXp-Span-869" class="MJXp-mo MJXp-script">∗</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-172-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1060" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1061" class="mrow"><span id="MathJax-Span-1062" class="msubsup"><span style="display: inline-block; position: relative; width: 1.056em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.59em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1063" class="mi">R</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1064" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-172" type="math/tex">R^*</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-870" class="MJXp-math"><span id="MJXp-Span-871" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-872" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-873" class="MJXp-mn MJXp-script">1</span></span><span id="MJXp-Span-874" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-875" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: -0.4em;" id="MJXp-Span-876" class="MJXp-mn MJXp-script">0</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-173-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1065" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1066" class="mrow"><span id="MathJax-Span-1067" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1068" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1069" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1070" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1071" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1072" class="mn">0</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-173" type="math/tex">u_1r_0</script></span> is the identity of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-877" class="MJXp-math"><span id="MJXp-Span-878" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-879" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-880" class="MJXp-mo">∗</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-882" class="MJXp-mi MJXp-italic">t</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-881" class="MJXp-mi MJXp-italic">a</span></span></span></span></span></span><span id="MJXp-Span-883" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-174-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1073" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1074" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1075" class="mi">R</span><span style="padding-left: 0.25em;" id="MathJax-Span-1076" class="msubsup"><span style="display: inline-block; position: relative; width: 0.952em; height: 0px;"><span style="position: absolute; clip: rect(3.375em, 1000.46em, 4.102em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-1077" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.46em, 1000.28em, 4.142em, -1000em); top: -4.343em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1078" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.013em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.142em, -1000em); top: -3.843em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1079" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1080" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-174" type="math/tex">R*_a^t G</script></span> and the map <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-884" class="MJXp-math"><span id="MJXp-Span-885" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-886" class="MJXp-mo">↦</span><span id="MJXp-Span-887" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-888" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-889" class="MJXp-mn MJXp-script">1</span></span><span id="MJXp-Span-890" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-891" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: -0.4em;" id="MJXp-Span-892" class="MJXp-mn MJXp-script">0</span></span><span id="MJXp-Span-893" class="MJXp-mi MJXp-italic">r</span></span></span><span style="" tabindex="0" id="MathJax-Element-175-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1081" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1082" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1083" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1084" class="mo">↦</span><span style="padding-left: 0.313em;" id="MathJax-Span-1085" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1086" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1087" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1088" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1089" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1090" class="mn">0</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1091" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-175" type="math/tex">r \mapsto u_1 r_0 r </script></span> is a ring isomorphism <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-894" class="MJXp-math"><span id="MJXp-Span-895" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-896" class="MJXp-mo">→</span><span id="MJXp-Span-897" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-898" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-899" class="MJXp-mn MJXp-script">1</span></span><span id="MJXp-Span-900" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-176-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1092" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1093" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1094" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1095" class="mo">→</span><span style="padding-left: 0.313em;" id="MathJax-Span-1096" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1097" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1098" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1099" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-176" type="math/tex">R \rightarrow u_1R </script></span>.</p>
-
-<p>Conversely, let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-901" class="MJXp-math"><span id="MJXp-Span-902" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-903" class="MJXp-mo">∗</span><span id="MJXp-Span-904" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-905" class="MJXp-mo">=</span><span id="MJXp-Span-906" class="MJXp-msubsup"><span style="margin-left: 0.111em; margin-right: 0.05em;" id="MJXp-Span-907" class="MJXp-mo">â¨</span><span style="vertical-align: -0.4em;" id="MJXp-Span-908" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-909" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-910" class="MJXp-mo">∈</span><span id="MJXp-Span-911" class="MJXp-mi MJXp-italic">G</span></span></span><span id="MJXp-Span-912" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-913" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-914" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-177-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1100" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1101" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1102" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1103" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1104" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1105" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1106" class="munderover"><span style="display: inline-block; position: relative; width: 2.549em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1001.07em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-1107" class="mo">â¨</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.694em; left: 1.126em;"><span id="MathJax-Span-1108" class="texatom"><span id="MathJax-Span-1109" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1110" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1111" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1112" class="mi">G</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-1113" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1114" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1115" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-177" type="math/tex">R*G=\bigoplus_{g\in G} A_g</script></span> be an (intrinsic) crossed product and select for each <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-915" class="MJXp-math"><span id="MJXp-Span-916" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-917" class="MJXp-mo">∈</span><span id="MJXp-Span-918" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-178-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1116" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1117" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1118" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1119" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1120" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-178" type="math/tex">g\in G</script></span> a unit <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-919" class="MJXp-math"><span id="MJXp-Span-920" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-921" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-922" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-923" class="MJXp-mo">∈</span><span id="MJXp-Span-924" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-925" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-926" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-179-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1121" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1122" class="mrow"><span id="MathJax-Span-1123" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1124" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1125" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1126" class="mo">∈</span><span style="padding-left: 0.313em;" id="MathJax-Span-1127" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1128" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1129" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-179" type="math/tex">u_g\in A_g</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-927" class="MJXp-math"><span id="MJXp-Span-928" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-929" class="MJXp-mo">∗</span><span id="MJXp-Span-930" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-180-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1130" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1131" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1132" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1133" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1134" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-180" type="math/tex">R*G</script></span>. This is called a <em>basis of units for the crossed product</em> <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-931" class="MJXp-math"><span id="MJXp-Span-932" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-933" class="MJXp-mo">∗</span><span id="MJXp-Span-934" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-181-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1135" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1136" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1137" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1138" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1139" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-181" type="math/tex">R*G</script></span>. Then the maps <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-935" class="MJXp-math"><span id="MJXp-Span-936" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-937" class="MJXp-mo">:</span><span id="MJXp-Span-938" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-939" class="MJXp-mo">→</span><span id="MJXp-Span-940" class="MJXp-mi MJXp-italic">A</span><span id="MJXp-Span-941" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-942" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-943" class="MJXp-mo">(</span><span id="MJXp-Span-944" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-945" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-182-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1140" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1141" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1142" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1143" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1144" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1145" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1146" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1147" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1148" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1149" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1150" class="mi">R</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1151" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-182" type="math/tex">a:G \rightarrow Aut(R)</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-946" class="MJXp-math"><span id="MJXp-Span-947" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-948" class="MJXp-mo">:</span><span id="MJXp-Span-949" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-950" class="MJXp-mo">×</span><span id="MJXp-Span-951" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-952" class="MJXp-mo">→</span><span id="MJXp-Span-953" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-954" class="MJXp-mi MJXp-italic">R</span><span style="vertical-align: 0.5em;" id="MJXp-Span-955" class="MJXp-mo MJXp-script">∗</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-183-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1152" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1153" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1154" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1155" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1156" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1157" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1158" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1159" class="mo">→</span><span style="padding-left: 0.313em;" id="MathJax-Span-1160" class="msubsup"><span style="display: inline-block; position: relative; width: 1.056em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.59em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1161" class="mi">R</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1162" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-183" type="math/tex">t:G\times G \rightarrow R^*</script></span> given by</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-956" class="MJXp-math MJXp-display"><span id="MJXp-Span-957" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-958" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: 0.5em;" id="MJXp-Span-959" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-960" class="MJXp-mo">=</span><span id="MJXp-Span-961" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-962" class="MJXp-mi MJXp-italic">u</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-964" class="MJXp-mrow"><span id="MJXp-Span-965" class="MJXp-mo">−</span><span id="MJXp-Span-966" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-963" class="MJXp-mi MJXp-italic">g</span></span></span></span></span></span><span id="MJXp-Span-967" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-968" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-969" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-970" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-971" class="MJXp-mo">,</span><span style="width: 1em; height: 0em;" id="MJXp-Span-972" class="MJXp-mspace"></span><span id="MJXp-Span-973" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-974" class="MJXp-mo">(</span><span id="MJXp-Span-975" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-976" class="MJXp-mo">,</span><span id="MJXp-Span-977" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-978" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-979" class="MJXp-mo">=</span><span id="MJXp-Span-980" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-981" class="MJXp-mi MJXp-italic">u</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-985" class="MJXp-mrow"><span id="MJXp-Span-986" class="MJXp-mo">−</span><span id="MJXp-Span-987" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-982" class="MJXp-mrow"><span id="MJXp-Span-983" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-984" class="MJXp-mi MJXp-italic">h</span></span></span></span></span></span></span><span id="MJXp-Span-988" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-989" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-990" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span id="MJXp-Span-991" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-992" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-993" class="MJXp-mi MJXp-italic MJXp-script">h</span></span><span style="width: 1em; height: 0em;" id="MJXp-Span-994" class="MJXp-mspace"></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-995" class="MJXp-mo">(</span><span id="MJXp-Span-996" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-997" class="MJXp-mo">,</span><span id="MJXp-Span-998" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-999" class="MJXp-mo">∈</span><span id="MJXp-Span-1000" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1001" class="MJXp-mo">,</span><span id="MJXp-Span-1002" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1003" class="MJXp-mo">∈</span><span id="MJXp-Span-1004" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1005" class="MJXp-mo">)</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-184-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-1163" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1164" class="mrow"><span id="MathJax-Span-1165" class="msubsup"><span style="display: inline-block; position: relative; width: 0.897em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1166" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.469em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1167" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1168" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1169" class="msubsup"><span style="display: inline-block; position: relative; width: 1.413em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1170" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.91em, 4.135em, -1000em); top: -4.335em; left: 0.5em;"><span id="MathJax-Span-1171" class="texatom"><span id="MathJax-Span-1172" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1173" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1174" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.28em, -1000em); top: -3.843em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1175" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1176" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span id="MathJax-Span-1177" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1178" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1179" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1180" class="mo">,</span><span style="height: 0em; vertical-align: 0em; width: 1.125em; display: inline-block; overflow: hidden;" id="MathJax-Span-1181" class="mspace"></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1182" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1183" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1184" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1185" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1186" class="mi">h</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1187" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1188" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1189" class="msubsup"><span style="display: inline-block; position: relative; width: 1.413em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1190" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.91em, 4.135em, -1000em); top: -4.335em; left: 0.5em;"><span id="MathJax-Span-1191" class="texatom"><span id="MathJax-Span-1192" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1193" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1194" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.78em, 4.28em, -1000em); top: -3.672em; left: 0.5em;"><span id="MathJax-Span-1195" class="texatom"><span id="MathJax-Span-1196" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1197" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1198" class="mi">h</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1199" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1200" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1201" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1202" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1203" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1204" class="mi">h</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="height: 0em; vertical-align: 0em; width: 1.125em; display: inline-block; overflow: hidden;" id="MathJax-Span-1205" class="mspace"></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1206" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1207" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1208" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1209" class="mi">h</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1210" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1211" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1212" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1213" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1214" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1215" class="mi">R</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1216" class="mo">)</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-184" type="math/tex; mode=display">
- r^g = u_g^{-1} r u_g, \quad t(g,h) = u_{gh}^{-1} u_g u_h \quad (g,h \in G, r \in R)
- </script></p>
-
-<p>satisfy conditions (1) and (2) and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1006" class="MJXp-math"><span id="MJXp-Span-1007" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1008" class="MJXp-mo">∗</span><span id="MJXp-Span-1009" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1010" class="MJXp-mo">=</span><span id="MJXp-Span-1011" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-1012" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-1013" class="MJXp-mo">∗</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-1015" class="MJXp-mi MJXp-italic">t</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1014" class="MJXp-mi MJXp-italic">a</span></span></span></span></span></span><span id="MJXp-Span-1016" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-185-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1217" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1218" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1219" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1220" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1221" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1222" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1223" class="mi">R</span><span style="padding-left: 0.25em;" id="MathJax-Span-1224" class="msubsup"><span style="display: inline-block; position: relative; width: 0.952em; height: 0px;"><span style="position: absolute; clip: rect(3.375em, 1000.46em, 4.102em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-1225" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.46em, 1000.28em, 4.142em, -1000em); top: -4.343em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1226" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.013em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.142em, -1000em); top: -3.843em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1227" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1228" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-185" type="math/tex">R*G = R*_a^t G</script></span>.</p>
-
-<p>The choice of a basis of units <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1017" class="MJXp-math"><span id="MJXp-Span-1018" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1019" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1020" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1021" class="MJXp-mo">∈</span><span id="MJXp-Span-1022" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1023" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1024" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-186-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1229" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1230" class="mrow"><span id="MathJax-Span-1231" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1232" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1233" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1234" class="mo">∈</span><span style="padding-left: 0.313em;" id="MathJax-Span-1235" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1236" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1237" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-186" type="math/tex">u_g \in A_g</script></span> determines the action <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1025" class="MJXp-math"><span id="MJXp-Span-1026" class="MJXp-mi MJXp-italic">a</span></span></span><span style="" tabindex="0" id="MathJax-Element-187-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1238" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1239" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1240" class="mi">a</span></span></span></span></span></nobr></span><script id="MathJax-Element-187" type="math/tex">a</script></span> and twisting <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1027" class="MJXp-math"><span id="MJXp-Span-1028" class="MJXp-mi MJXp-italic">t</span></span></span><span style="" tabindex="0" id="MathJax-Element-188-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1241" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1242" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1243" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-188" type="math/tex">t</script></span>. If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1029" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1030" class="MJXp-mo">{</span><span id="MJXp-Span-1031" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1032" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1033" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1034" class="MJXp-mo">∈</span><span id="MJXp-Span-1035" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1036" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1037" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-1038" class="MJXp-mo">:</span><span id="MJXp-Span-1039" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1040" class="MJXp-mo">∈</span><span id="MJXp-Span-1041" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1042" class="MJXp-mo">}</span></span></span><span style="" tabindex="0" id="MathJax-Element-189-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1244" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1245" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1246" class="mo">{</span><span id="MathJax-Span-1247" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1248" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1249" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1250" class="mo">∈</span><span style="padding-left: 0.313em;" id="MathJax-Span-1251" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1252" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1253" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1254" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1255" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1256" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1257" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1258" class="mo">}</span></span></span></span></span></nobr></span><script id="MathJax-Element-189" type="math/tex">\{u_g \in A_g : g \in G \}</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1043" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1044" class="MJXp-mo">{</span><span id="MJXp-Span-1045" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1046" class="MJXp-mi MJXp-italic">v</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1047" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1048" class="MJXp-mo">∈</span><span id="MJXp-Span-1049" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1050" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1051" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-1052" class="MJXp-mo">:</span><span id="MJXp-Span-1053" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1054" class="MJXp-mo">∈</span><span id="MJXp-Span-1055" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1056" class="MJXp-mo">}</span></span></span><span style="" tabindex="0" id="MathJax-Element-190-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1259" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1260" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1261" class="mo">{</span><span id="MathJax-Span-1262" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.43em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1263" class="mi">v</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1264" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1265" class="mo">∈</span><span style="padding-left: 0.313em;" id="MathJax-Span-1266" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1267" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1268" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1269" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1270" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1271" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1272" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1273" class="mo">}</span></span></span></span></span></nobr></span><script id="MathJax-Element-190" type="math/tex">\{v_g \in A_g : g \in G \}</script></span> are two sets of units of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1057" class="MJXp-math"><span id="MJXp-Span-1058" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1059" class="MJXp-mo">∗</span><span id="MJXp-Span-1060" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-191-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1274" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1275" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1276" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1277" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1278" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-191" type="math/tex">R*G</script></span> then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1061" class="MJXp-math"><span id="MJXp-Span-1062" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1063" class="MJXp-mi MJXp-italic">v</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1064" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1065" class="MJXp-mo">=</span><span id="MJXp-Span-1066" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1067" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1068" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span id="MJXp-Span-1069" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1070" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1071" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-192-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1279" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1280" class="mrow"><span id="MathJax-Span-1281" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.43em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1282" class="mi">v</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1283" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1284" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1285" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1286" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1287" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1288" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1289" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1290" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-192" type="math/tex">v_g = u_g r_g</script></span> for some units <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1072" class="MJXp-math"><span id="MJXp-Span-1073" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1074" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1075" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-193-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1291" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1292" class="mrow"><span id="MathJax-Span-1293" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1294" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1295" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-193" type="math/tex">r_g</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1076" class="MJXp-math"><span id="MJXp-Span-1077" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-194-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1296" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1297" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1298" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-194" type="math/tex">R</script></span>.
- Changing the basis of units results in a change of the action and the
-twisting and so changes the extrinsic definition of the crossed product
-but it does not change the intrinsic crossed product.</p>
-
-<p>It is customary to select <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1078" class="MJXp-math"><span id="MJXp-Span-1079" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1080" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1081" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1082" class="MJXp-mo">=</span><span id="MJXp-Span-1083" class="MJXp-mn">1</span></span></span><span style="" tabindex="0" id="MathJax-Element-195-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1299" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1300" class="mrow"><span id="MathJax-Span-1301" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1302" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1303" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1304" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1305" class="mn">1</span></span></span></span></span></nobr></span><script id="MathJax-Element-195" type="math/tex">u_1=1</script></span>. In that case <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1084" class="MJXp-math"><span id="MJXp-Span-1085" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1086" class="MJXp-mo">(</span><span id="MJXp-Span-1087" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1088" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-196-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1306" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1307" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1308" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1309" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1310" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1311" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-196" type="math/tex">a(1)</script></span> is the identity map of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1089" class="MJXp-math"><span id="MJXp-Span-1090" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-197-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1312" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1313" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1314" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-197" type="math/tex">R</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1091" class="MJXp-math"><span id="MJXp-Span-1092" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1093" class="MJXp-mo">(</span><span id="MJXp-Span-1094" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1095" class="MJXp-mo">,</span><span id="MJXp-Span-1096" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1097" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1098" class="MJXp-mo">=</span><span id="MJXp-Span-1099" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1100" class="MJXp-mo">(</span><span id="MJXp-Span-1101" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1102" class="MJXp-mo">,</span><span id="MJXp-Span-1103" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1104" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1105" class="MJXp-mo">=</span><span id="MJXp-Span-1106" class="MJXp-mn">1</span></span></span><span style="" tabindex="0" id="MathJax-Element-198-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1315" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1316" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1317" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1318" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1319" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1320" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1321" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1322" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1323" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1324" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1325" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1326" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1327" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1328" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1329" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1330" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1331" class="mn">1</span></span></span></span></span></nobr></span><script id="MathJax-Element-198" type="math/tex">t(1,g)=t(g,1)=1</script></span> for each <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1107" class="MJXp-math"><span id="MJXp-Span-1108" class="MJXp-mi MJXp-italic">g</span></span></span><span style="" tabindex="0" id="MathJax-Element-199-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1332" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1333" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1334" class="mi">g</span></span></span></span></span></nobr></span><script id="MathJax-Element-199" type="math/tex">g</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1109" class="MJXp-math"><span id="MJXp-Span-1110" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-200-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1335" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1336" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1337" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-200" type="math/tex">G</script></span>.</p>
-
-<p><a id="X828C42CD86AF605F" name="X828C42CD86AF605F"></a></p>
-
-<h4>9.7 <span class="Heading">Cyclic Crossed Products</span></h4>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1111" class="MJXp-math"><span id="MJXp-Span-1112" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1113" class="MJXp-mo">∗</span><span id="MJXp-Span-1114" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1115" class="MJXp-mo">=</span><span id="MJXp-Span-1116" class="MJXp-msubsup"><span style="margin-left: 0.111em; margin-right: 0.05em;" id="MJXp-Span-1117" class="MJXp-mo">â¨</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1118" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1119" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-1120" class="MJXp-mo">∈</span><span id="MJXp-Span-1121" class="MJXp-mi MJXp-italic">G</span></span></span><span id="MJXp-Span-1122" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1123" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1124" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-201-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1338" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1339" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1340" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1341" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1342" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1343" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1344" class="munderover"><span style="display: inline-block; position: relative; width: 2.549em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1001.07em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-1345" class="mo">â¨</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.694em; left: 1.126em;"><span id="MathJax-Span-1346" class="texatom"><span id="MathJax-Span-1347" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1348" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1349" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1350" class="mi">G</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-1351" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1352" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1353" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-201" type="math/tex">R*G=\bigoplus_{g \in G} A_g</script></span> be a <em>crossed product</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7FB21779832CE1CB"><span class="RefLink">9.6</span></a>) and assume that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1125" class="MJXp-math"><span id="MJXp-Span-1126" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1127" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1128" class="MJXp-mo">⟨</span><span id="MJXp-Span-1129" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1130" class="MJXp-mo">⟩</span></span></span><span style="" tabindex="0" id="MathJax-Element-202-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1354" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1355" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1356" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1357" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1358" class="mo">⟨</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1359" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1360" class="mo">⟩</span></span></span></span></span></nobr></span><script id="MathJax-Element-202" type="math/tex">G = \langle g \rangle </script></span> is cyclic. Then the crossed product can be given using a particularly nice description.</p>
-
-<p>Select a unit <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1131" class="MJXp-math"><span id="MJXp-Span-1132" class="MJXp-mi MJXp-italic">u</span></span></span><span style="" tabindex="0" id="MathJax-Element-203-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1361" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1362" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1363" class="mi">u</span></span></span></span></span></nobr></span><script id="MathJax-Element-203" type="math/tex">u</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1133" class="MJXp-math"><span id="MJXp-Span-1134" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1135" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1136" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1137" class="MJXp-mi MJXp-italic">g</span></span></span></span></span><span style="" tabindex="0" id="MathJax-Element-204-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1364" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1365" class="mrow"><span id="MathJax-Span-1366" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1367" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span id="MathJax-Span-1368" class="texatom"><span id="MathJax-Span-1369" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1370" class="mi">g</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-204" type="math/tex">A_{g}</script></span>, and let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1138" class="MJXp-math"><span id="MJXp-Span-1139" class="MJXp-mi MJXp-italic">a</span></span></span><span style="" tabindex="0" id="MathJax-Element-205-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1371" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1372" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1373" class="mi">a</span></span></span></span></span></nobr></span><script id="MathJax-Element-205" type="math/tex">a</script></span> be the automorphism of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1140" class="MJXp-math"><span id="MJXp-Span-1141" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-206-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1374" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1375" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1376" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-206" type="math/tex">R</script></span> given by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1142" class="MJXp-math"><span id="MJXp-Span-1143" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1144" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1145" class="MJXp-mi MJXp-italic MJXp-script">a</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1146" class="MJXp-mo">=</span><span id="MJXp-Span-1147" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1148" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1149" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1150" class="MJXp-mo">−</span><span id="MJXp-Span-1151" class="MJXp-mn">1</span></span></span><span id="MJXp-Span-1152" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-1153" class="MJXp-mi MJXp-italic">u</span></span></span><span style="" tabindex="0" id="MathJax-Element-207-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1377" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1378" class="mrow"><span id="MathJax-Span-1379" class="msubsup"><span style="display: inline-block; position: relative; width: 0.898em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1380" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.469em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1381" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1382" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1383" class="msubsup"><span style="display: inline-block; position: relative; width: 1.413em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1384" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.5em;"><span id="MathJax-Span-1385" class="texatom"><span id="MathJax-Span-1386" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1387" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1388" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1389" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1390" class="mi">u</span></span></span></span></span></nobr></span><script id="MathJax-Element-207" type="math/tex">r^a = u^{-1} r u</script></span>.</p>
-
-<p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1154" class="MJXp-math"><span id="MJXp-Span-1155" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-208-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1391" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1392" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1393" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-208" type="math/tex">G</script></span> is infinite then set <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1156" class="MJXp-math"><span id="MJXp-Span-1157" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1158" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1159" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1160" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1161" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1162" class="MJXp-mi MJXp-italic MJXp-script">k</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1163" class="MJXp-mo">=</span><span id="MJXp-Span-1164" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1165" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1166" class="MJXp-mi MJXp-italic MJXp-script">k</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-209-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1394" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1395" class="mrow"><span id="MathJax-Span-1396" class="msubsup"><span style="display: inline-block; position: relative; width: 1.212em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1397" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.789em; left: 0.5em;"><span id="MathJax-Span-1398" class="texatom"><span id="MathJax-Span-1399" class="mrow"><span id="MathJax-Span-1400" class="msubsup"><span style="display: inline-block; position: relative; width: 0.637em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.33em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1401" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.195em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1402" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.009em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1403" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1404" class="msubsup"><span style="display: inline-block; position: relative; width: 0.901em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1405" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1406" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-209" type="math/tex">u_{g^k} = u^k</script></span> for every integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1167" class="MJXp-math"><span id="MJXp-Span-1168" class="MJXp-mi MJXp-italic">k</span></span></span><span style="" tabindex="0" id="MathJax-Element-210-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1407" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1408" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1409" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-210" type="math/tex">k</script></span>. Then</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1169" class="MJXp-math MJXp-display"><span id="MJXp-Span-1170" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1171" class="MJXp-mo">∗</span><span id="MJXp-Span-1172" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1173" class="MJXp-mo">=</span><span id="MJXp-Span-1174" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1175" class="MJXp-mo">[</span><span id="MJXp-Span-1176" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-1177" class="MJXp-mrow"><span style="margin-left: 0.167em; margin-right: 0.167em;" id="MJXp-Span-1178" class="MJXp-mo">|</span></span><span id="MJXp-Span-1179" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-1180" class="MJXp-mi MJXp-italic">u</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1181" class="MJXp-mo">=</span><span id="MJXp-Span-1182" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-1183" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1184" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1185" class="MJXp-mi MJXp-italic MJXp-script">a</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1186" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1187" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-211-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-1410" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1411" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1412" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1413" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1414" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1415" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1416" class="mi">R</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1417" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1418" class="mi">u</span><span id="MathJax-Span-1419" class="texatom"><span id="MathJax-Span-1420" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1421" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1422" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1423" class="mi">u</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1424" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1425" class="mi">u</span><span id="MathJax-Span-1426" class="msubsup"><span style="display: inline-block; position: relative; width: 0.898em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1427" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.469em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1428" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1429" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1430" class="mo">,</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-211" type="math/tex; mode=display">
- R*G = R[ u | ru = u r^a ],
- </script></p>
-
-<p>a skew polynomial ring. Therefore in this case <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1188" class="MJXp-math"><span id="MJXp-Span-1189" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1190" class="MJXp-mo">∗</span><span id="MJXp-Span-1191" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-212-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1431" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1432" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1433" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1434" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1435" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-212" type="math/tex">R*G</script></span> is determined by</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1192" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1193" class="MJXp-mo">[</span><span id="MJXp-Span-1194" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1195" class="MJXp-mo">,</span><span id="MJXp-Span-1196" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1197" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1198" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-213-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-1436" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1437" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1438" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1439" class="mi">R</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1440" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1441" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1442" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1443" class="mo">.</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-213" type="math/tex; mode=display">
-[ R, a ].
-</script></p>
-
-<p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1199" class="MJXp-math"><span id="MJXp-Span-1200" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-214-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1444" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1445" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1446" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-214" type="math/tex">G</script></span> is finite of order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1201" class="MJXp-math"><span id="MJXp-Span-1202" class="MJXp-mi MJXp-italic">d</span></span></span><span style="" tabindex="0" id="MathJax-Element-215-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1447" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1448" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1449" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-215" type="math/tex">d</script></span> then set <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1203" class="MJXp-math"><span id="MJXp-Span-1204" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1205" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1206" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1207" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1208" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1209" class="MJXp-mi MJXp-italic MJXp-script">k</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1210" class="MJXp-mo">=</span><span id="MJXp-Span-1211" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1212" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1213" class="MJXp-mi MJXp-italic MJXp-script">k</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-216-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1450" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1451" class="mrow"><span id="MathJax-Span-1452" class="msubsup"><span style="display: inline-block; position: relative; width: 1.212em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1453" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.789em; left: 0.5em;"><span id="MathJax-Span-1454" class="texatom"><span id="MathJax-Span-1455" class="mrow"><span id="MathJax-Span-1456" class="msubsup"><span style="display: inline-block; position: relative; width: 0.637em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.33em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1457" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.195em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1458" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.009em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1459" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1460" class="msubsup"><span style="display: inline-block; position: relative; width: 0.901em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1461" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1462" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-216" type="math/tex">u_{g^k} = u^k</script></span> for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1214" class="MJXp-math"><span id="MJXp-Span-1215" class="MJXp-mn">0</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1216" class="MJXp-mo">≤</span><span id="MJXp-Span-1217" class="MJXp-mi MJXp-italic">k</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1218" class="MJXp-mo"><</span><span id="MJXp-Span-1219" class="MJXp-mi MJXp-italic">d</span></span></span><span style="" tabindex="0" id="MathJax-Element-217-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1463" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1464" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1465" class="mn">0</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1466" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1467" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1468" class="mo"><</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1469" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-217" type="math/tex">0 \le k < d</script></span>. Then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1220" class="MJXp-math"><span id="MJXp-Span-1221" class="MJXp-mi MJXp-italic">b</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1222" class="MJXp-mo">=</span><span id="MJXp-Span-1223" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1224" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1225" class="MJXp-mi MJXp-italic MJXp-script">d</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1226" class="MJXp-mo">∈</span><span id="MJXp-Span-1227" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-218-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1470" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1471" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1472" class="mi">b</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1473" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1474" class="msubsup"><span style="display: inline-block; position: relative; width: 0.948em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1475" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1476" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1477" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1478" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-218" type="math/tex"> b = u^d \in R </script></span> and</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1228" class="MJXp-math MJXp-display"><span id="MJXp-Span-1229" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1230" class="MJXp-mo">∗</span><span id="MJXp-Span-1231" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1232" class="MJXp-mo">=</span><span id="MJXp-Span-1233" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1234" class="MJXp-mo">[</span><span id="MJXp-Span-1235" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-1236" class="MJXp-mrow"><span style="margin-left: 0.167em; margin-right: 0.167em;" id="MJXp-Span-1237" class="MJXp-mo">|</span></span><span id="MJXp-Span-1238" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-1239" class="MJXp-mi MJXp-italic">u</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1240" class="MJXp-mo">=</span><span id="MJXp-Span-1241" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-1242" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1243" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1244" class="MJXp-mi MJXp-italic MJXp-script">a</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1245" class="MJXp-mo">,</span><span id="MJXp-Span-1246" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1247" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1248" class="MJXp-mi MJXp-italic MJXp-script">d</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1249" class="MJXp-mo">=</span><span id="MJXp-Span-1250" class="MJXp-mi MJXp-italic">b</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1251" class="MJXp-mo">]</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-219-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-1479" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1480" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1481" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1482" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1483" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1484" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1485" class="mi">R</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1486" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1487" class="mi">u</span><span id="MathJax-Span-1488" class="texatom"><span id="MathJax-Span-1489" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1490" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1491" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1492" class="mi">u</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1493" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1494" class="mi">u</span><span id="MathJax-Span-1495" class="msubsup"><span style="display: inline-block; position: relative; width: 0.898em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1496" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.469em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1497" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1498" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1499" class="msubsup"><span style="display: inline-block; position: relative; width: 0.948em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1500" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1501" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1502" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1503" class="mi">b</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1504" class="mo">]</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-219" type="math/tex; mode=display">
- R*G = R[ u | ru = u r^a, u^d = b ]
- </script></p>
-
-<p>Therefore, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1252" class="MJXp-math"><span id="MJXp-Span-1253" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1254" class="MJXp-mo">∗</span><span id="MJXp-Span-1255" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-220-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1505" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1506" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1507" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1508" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1509" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-220" type="math/tex">R*G</script></span> is completely determined by the following data:</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1256" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1257" class="MJXp-mo">[</span><span id="MJXp-Span-1258" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1259" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1260" class="MJXp-mo">[</span><span id="MJXp-Span-1261" class="MJXp-mi MJXp-italic">d</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1262" class="MJXp-mo">,</span><span id="MJXp-Span-1263" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1264" class="MJXp-mo">,</span><span id="MJXp-Span-1265" class="MJXp-mi MJXp-italic">b</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1266" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1267" class="MJXp-mo">]</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-221-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-1510" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1511" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1512" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1513" class="mi">R</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1514" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1515" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1516" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1517" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1518" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1519" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1520" class="mi">b</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1521" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1522" class="mo">]</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-221" type="math/tex; mode=display">
- [ R , [ d , a , b ] ]
- </script></p>
-
-<p><a id="X7869E2A48784C232" name="X7869E2A48784C232"></a></p>
-
-<h4>9.8 <span class="Heading">Abelian Crossed Products</span></h4>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1268" class="MJXp-math"><span id="MJXp-Span-1269" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1270" class="MJXp-mo">∗</span><span id="MJXp-Span-1271" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1272" class="MJXp-mo">=</span><span id="MJXp-Span-1273" class="MJXp-msubsup"><span style="margin-left: 0.111em; margin-right: 0.05em;" id="MJXp-Span-1274" class="MJXp-mo">â¨</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1275" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1276" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-1277" class="MJXp-mo">∈</span><span id="MJXp-Span-1278" class="MJXp-mi MJXp-italic">G</span></span></span><span id="MJXp-Span-1279" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1280" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1281" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-222-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1523" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1524" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1525" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1526" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1527" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1528" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1529" class="munderover"><span style="display: inline-block; position: relative; width: 2.549em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1001.07em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-1530" class="mo">â¨</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.694em; left: 1.126em;"><span id="MathJax-Span-1531" class="texatom"><span id="MathJax-Span-1532" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1533" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1534" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1535" class="mi">G</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-1536" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1537" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1538" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-222" type="math/tex">R*G=\bigoplus_{g \in G} A_g</script></span> be a <em>crossed product</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7FB21779832CE1CB"><span class="RefLink">9.6</span></a>) and assume that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1282" class="MJXp-math"><span id="MJXp-Span-1283" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-223-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1539" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1540" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1541" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-223" type="math/tex">G</script></span> is abelian. Then the crossed product can be given using a simple description.</p>
-
-<p>Express <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1284" class="MJXp-math"><span id="MJXp-Span-1285" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-224-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1542" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1543" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1544" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-224" type="math/tex">G</script></span> as a direct sum of cyclic groups:</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1286" class="MJXp-math MJXp-display"><span id="MJXp-Span-1287" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1288" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1289" class="MJXp-mo">⟨</span><span id="MJXp-Span-1290" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1291" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1292" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1293" class="MJXp-mo">⟩</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1294" class="MJXp-mo">×</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1295" class="MJXp-mo">⋯</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1296" class="MJXp-mo">×</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1297" class="MJXp-mo">⟨</span><span id="MJXp-Span-1298" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1299" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1300" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1301" class="MJXp-mo">⟩</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-225-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-1545" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1546" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1547" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1548" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1549" class="mo">⟨</span><span id="MathJax-Span-1550" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1551" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1552" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1553" class="mo">⟩</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1554" class="mo">×</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1555" class="mo">⋯</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1556" class="mo">×</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1557" class="mo">⟨</span><span id="MathJax-Span-1558" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1559" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1560" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1561" class="mo">⟩</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-225" type="math/tex; mode=display">
- G = \langle g_1 \rangle \times \cdots \times \langle g_n \rangle
- </script></p>
-
-<p>and for each <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1302" class="MJXp-math"><span id="MJXp-Span-1303" class="MJXp-mi MJXp-italic">i</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1304" class="MJXp-mo">=</span><span id="MJXp-Span-1305" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1306" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1307" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1308" class="MJXp-mo">,</span><span id="MJXp-Span-1309" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-226-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1562" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1563" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1564" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1565" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1566" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1567" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1568" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1569" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1570" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-226" type="math/tex">i=1,\dots,n</script></span> select a unit <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1310" class="MJXp-math"><span id="MJXp-Span-1311" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1312" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1313" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-227-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1571" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1572" class="mrow"><span id="MathJax-Span-1573" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1574" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1575" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-227" type="math/tex">u_i</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1314" class="MJXp-math"><span id="MJXp-Span-1315" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1316" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1317" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1318" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1319" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1320" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span></span></span><span style="" tabindex="0" id="MathJax-Element-228-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1576" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1577" class="mrow"><span id="MathJax-Span-1578" class="msubsup"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1579" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span id="MathJax-Span-1580" class="texatom"><span id="MathJax-Span-1581" class="mrow"><span id="MathJax-Span-1582" class="msubsup"><span style="display: inline-block; position: relative; width: 0.546em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.33em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1583" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.82em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1584" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-228" type="math/tex">A_{g_i}</script></span>.</p>
-
-<p>Each element <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1321" class="MJXp-math"><span id="MJXp-Span-1322" class="MJXp-mi MJXp-italic">g</span></span></span><span style="" tabindex="0" id="MathJax-Element-229-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1585" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1586" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1587" class="mi">g</span></span></span></span></span></nobr></span><script id="MathJax-Element-229" type="math/tex">g</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1323" class="MJXp-math"><span id="MJXp-Span-1324" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-230-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1588" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1589" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1590" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-230" type="math/tex">G</script></span> has a unique expression</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1325" class="MJXp-math MJXp-display"><span id="MJXp-Span-1326" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1327" class="MJXp-mo">=</span><span id="MJXp-Span-1328" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1329" class="MJXp-mi MJXp-italic">g</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.59em;"><span id="MJXp-Span-1331" class="MJXp-mrow"><span id="MJXp-Span-1332" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1333" class="MJXp-mi MJXp-italic">k</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1334" class="MJXp-mn MJXp-script">1</span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1330" class="MJXp-mn">1</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1335" class="MJXp-mo">⋯</span><span id="MJXp-Span-1336" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1337" class="MJXp-mi MJXp-italic">g</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.59em;"><span id="MJXp-Span-1339" class="MJXp-mrow"><span id="MJXp-Span-1340" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1341" class="MJXp-mi MJXp-italic">k</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1342" class="MJXp-mi MJXp-italic MJXp-script">n</span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1338" class="MJXp-mi MJXp-italic">n</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1343" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-231-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-1591" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1592" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1593" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1594" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1595" class="msubsup"><span style="display: inline-block; position: relative; width: 1.192em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1596" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.69em, 4.241em, -1000em); top: -4.441em; left: 0.5em;"><span id="MathJax-Span-1597" class="texatom"><span id="MathJax-Span-1598" class="mrow"><span id="MathJax-Span-1599" class="msubsup"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.363em, 1000.33em, 4.142em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1600" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.314em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1601" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.43em, 4.135em, -1000em); top: -3.677em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1602" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1603" class="mo">⋯</span><span style="padding-left: 0.188em;" id="MathJax-Span-1604" class="msubsup"><span style="display: inline-block; position: relative; width: 1.192em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1605" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.69em, 4.245em, -1000em); top: -4.445em; left: 0.5em;"><span id="MathJax-Span-1606" class="texatom"><span id="MathJax-Span-1607" class="mrow"><span id="MathJax-Span-1608" class="msubsup"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.363em, 1000.33em, 4.142em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1609" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.314em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1610" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.141em, -1000em); top: -3.843em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1611" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1612" class="mo">,</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-231" type="math/tex; mode=display">
- g = g_1^{k_1} \cdots g_n^{k_n},
- </script></p>
-
-<p>where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1344" class="MJXp-math"><span id="MJXp-Span-1345" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1346" class="MJXp-mi MJXp-italic">k</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1347" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-232-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1613" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1614" class="mrow"><span id="MathJax-Span-1615" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.46em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1616" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1617" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-232" type="math/tex">k_i</script></span> is an arbitrary integer, if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1348" class="MJXp-math"><span id="MJXp-Span-1349" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1350" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1351" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-233-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1618" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1619" class="mrow"><span id="MathJax-Span-1620" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1621" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1622" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-233" type="math/tex">g_i</script></span> has infinite order, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1352" class="MJXp-math"><span id="MJXp-Span-1353" class="MJXp-mn">0</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1354" class="MJXp-mo">≤</span><span id="MJXp-Span-1355" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1356" class="MJXp-mi MJXp-italic">k</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1357" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1358" class="MJXp-mo"><</span><span id="MJXp-Span-1359" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1360" class="MJXp-mi MJXp-italic">d</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1361" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-234-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1623" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1624" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1625" class="mn">0</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1626" class="mo">≤</span><span style="padding-left: 0.313em;" id="MathJax-Span-1627" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.46em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1628" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1629" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1630" class="mo"><</span><span style="padding-left: 0.313em;" id="MathJax-Span-1631" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.53em, 4.148em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1632" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1633" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-234" type="math/tex">0 \le k_i < d_i</script></span>, if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1362" class="MJXp-math"><span id="MJXp-Span-1363" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1364" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1365" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-235-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1634" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1635" class="mrow"><span id="MathJax-Span-1636" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1637" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1638" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-235" type="math/tex">g_i</script></span> has finite order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1366" class="MJXp-math"><span id="MJXp-Span-1367" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1368" class="MJXp-mi MJXp-italic">d</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1369" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-236-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1639" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1640" class="mrow"><span id="MathJax-Span-1641" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.53em, 4.148em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1642" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1643" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-236" type="math/tex">d_i</script></span>. Then one selects a basis for the crossed product by taking</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1370" class="MJXp-math MJXp-display"><span id="MJXp-Span-1371" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1372" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1373" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1374" class="MJXp-mo">=</span><span id="MJXp-Span-1375" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1376" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1377" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1378" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1379" class="MJXp-mi MJXp-italic">g</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.59em;"><span id="MJXp-Span-1381" class="MJXp-mrow"><span id="MJXp-Span-1382" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1383" class="MJXp-mi MJXp-italic">k</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1384" class="MJXp-mn MJXp-script">1</span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1380" class="MJXp-mn">1</span></span></span></span></span></span><span id="MJXp-Span-1385" class="MJXp-mo">⋯</span><span id="MJXp-Span-1386" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1387" class="MJXp-mi MJXp-italic">g</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.59em;"><span id="MJXp-Span-1389" class="MJXp-mrow"><span id="MJXp-Span-1390" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1391" class="MJXp-mi MJXp-italic">k</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1392" class="MJXp-mi MJXp-italic MJXp-script">n</span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1388" class="MJXp-mi MJXp-italic">n</span></span></span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1393" class="MJXp-mo">=</span><span id="MJXp-Span-1394" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1395" class="MJXp-mi MJXp-italic">u</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.59em;"><span id="MJXp-Span-1397" class="MJXp-mrow"><span id="MJXp-Span-1398" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1399" class="MJXp-mi MJXp-italic">k</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1400" class="MJXp-mn MJXp-script">1</span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1396" class="MJXp-mn">1</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1401" class="MJXp-mo">⋯</span><span id="MJXp-Span-1402" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1403" class="MJXp-mi MJXp-italic">u</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.59em;"><span id="MJXp-Span-1405" class="MJXp-mrow"><span id="MJXp-Span-1406" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1407" class="MJXp-mi MJXp-italic">k</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1408" class="MJXp-mi MJXp-italic MJXp-script">n</span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1404" class="MJXp-mi MJXp-italic">n</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1409" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-237-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-1644" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1645" class="mrow"><span id="MathJax-Span-1646" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1647" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1648" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1649" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1650" class="msubsup"><span style="display: inline-block; position: relative; width: 3.062em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1651" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.584em; left: 0.5em;"><span id="MathJax-Span-1652" class="texatom"><span id="MathJax-Span-1653" class="mrow"><span id="MathJax-Span-1654" class="msubsup"><span style="display: inline-block; position: relative; width: 0.916em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.33em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1655" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.505em, 1000.56em, 4.3em, -1000em); top: -4.4em; left: 0.354em;"><span id="MathJax-Span-1656" class="texatom"><span id="MathJax-Span-1657" class="mrow"><span id="MathJax-Span-1658" class="msubsup"><span style="display: inline-block; position: relative; width: 0.51em; height: 0px;"><span style="position: absolute; clip: rect(3.505em, 1000.23em, 4.14em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1659" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.009em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.825em; left: 0.222em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1660" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.508em, 1000.3em, 4.135em, -1000em); top: -3.769em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1661" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1662" class="mo">⋯</span><span id="MathJax-Span-1663" class="msubsup"><span style="display: inline-block; position: relative; width: 0.916em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.33em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1664" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.505em, 1000.56em, 4.214em, -1000em); top: -4.313em; left: 0.354em;"><span id="MathJax-Span-1665" class="texatom"><span id="MathJax-Span-1666" class="mrow"><span id="MathJax-Span-1667" class="msubsup"><span style="display: inline-block; position: relative; width: 0.51em; height: 0px;"><span style="position: absolute; clip: rect(3.505em, 1000.23em, 4.14em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1668" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.009em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.915em; left: 0.222em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1669" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.626em, 1000.3em, 4.139em, -1000em); top: -3.886em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1670" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1671" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1672" class="msubsup"><span style="display: inline-block; position: relative; width: 1.192em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1673" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.69em, 4.241em, -1000em); top: -4.441em; left: 0.5em;"><span id="MathJax-Span-1674" class="texatom"><span id="MathJax-Span-1675" class="mrow"><span id="MathJax-Span-1676" class="msubsup"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.363em, 1000.33em, 4.142em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1677" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.314em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1678" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.43em, 4.135em, -1000em); top: -3.677em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1679" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1680" class="mo">⋯</span><span style="padding-left: 0.188em;" id="MathJax-Span-1681" class="msubsup"><span style="display: inline-block; position: relative; width: 1.192em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1682" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.69em, 4.245em, -1000em); top: -4.445em; left: 0.5em;"><span id="MathJax-Span-1683" class="texatom"><span id="MathJax-Span-1684" class="mrow"><span id="MathJax-Span-1685" class="msubsup"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.363em, 1000.33em, 4.142em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1686" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.314em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1687" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.141em, -1000em); top: -3.843em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1688" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1689" class="mo">.</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-237" type="math/tex; mode=display">
- u_g = u_{g_1^{k_1} \cdots g_n^{k_n}} = u_1^{k_1} \cdots u_n^{k_n}.
- </script></p>
-
-<p>* For each <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1410" class="MJXp-math"><span id="MJXp-Span-1411" class="MJXp-mi MJXp-italic">i</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1412" class="MJXp-mo">=</span><span id="MJXp-Span-1413" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1414" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1415" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1416" class="MJXp-mo">,</span><span id="MJXp-Span-1417" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-238-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1690" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1691" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1692" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1693" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1694" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1695" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1696" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1697" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1698" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-238" type="math/tex">i=1,\dots, n</script></span>, let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1418" class="MJXp-math"><span id="MJXp-Span-1419" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1420" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1421" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-239-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1699" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1700" class="mrow"><span id="MathJax-Span-1701" class="msubsup"><span style="display: inline-block; position: relative; width: 0.773em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1702" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.501em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1703" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-239" type="math/tex">a_i</script></span> be the automorphism of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1422" class="MJXp-math"><span id="MJXp-Span-1423" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-240-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1704" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1705" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1706" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-240" type="math/tex">R</script></span> given by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1424" class="MJXp-math"><span id="MJXp-Span-1425" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1426" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1427" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1428" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1429" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1430" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1431" class="MJXp-mo">=</span><span id="MJXp-Span-1432" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1433" class="MJXp-mi MJXp-italic">u</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-1435" class="MJXp-mrow"><span id="MJXp-Span-1436" class="MJXp-mo">−</span><span id="MJXp-Span-1437" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1434" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span><span id="MJXp-Span-1438" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-1439" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1440" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1441" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-241-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1707" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1708" class="mrow"><span id="MathJax-Span-1709" class="msubsup"><span style="display: inline-block; position: relative; width: 1.09em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1710" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.469em;"><span id="MathJax-Span-1711" class="texatom"><span id="MathJax-Span-1712" class="mrow"><span id="MathJax-Span-1713" class="msubsup"><span style="display: inline-block; position: relative; width: 0.546em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.142em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1714" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1715" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1716" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1717" class="msubsup"><span style="display: inline-block; position: relative; width: 1.413em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1718" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.91em, 4.135em, -1000em); top: -4.335em; left: 0.5em;"><span id="MathJax-Span-1719" class="texatom"><span id="MathJax-Span-1720" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1721" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1722" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.27em, 4.142em, -1000em); top: -3.692em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1723" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1724" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span id="MathJax-Span-1725" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1726" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1727" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-241" type="math/tex">r^{a_i} = u_i^{-1} r u_i</script></span>.</p>
-
-<p>* For each <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1442" class="MJXp-math"><span id="MJXp-Span-1443" class="MJXp-mn">1</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1444" class="MJXp-mo">≤</span><span id="MJXp-Span-1445" class="MJXp-mi MJXp-italic">i</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1446" class="MJXp-mo"><</span><span id="MJXp-Span-1447" class="MJXp-mi MJXp-italic">j</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1448" class="MJXp-mo">≤</span><span id="MJXp-Span-1449" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-242-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1728" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1729" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1730" class="mn">1</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1731" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1732" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1733" class="mo"><</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1734" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1735" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1736" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-242" type="math/tex">1 \le i < j \le n</script></span>, let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1450" class="MJXp-math"><span id="MJXp-Span-1451" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1452" class="MJXp-mi MJXp-italic">t</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1453" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1454" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-1455" class="MJXp-mo">,</span><span id="MJXp-Span-1456" class="MJXp-mi MJXp-italic">j</span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1457" class="MJXp-mo">=</span><span id="MJXp-Span-1458" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1459" class="MJXp-mi MJXp-italic">u</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-1461" class="MJXp-mrow"><span id="MJXp-Span-1462" class="MJXp-mo">−</span><span id="MJXp-Span-1463" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1460" class="MJXp-mi MJXp-italic">j</span></span></span></span></span></span><span id="MJXp-Span-1464" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1465" class="MJXp-mi MJXp-italic">u</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-1467" class="MJXp-mrow"><span id="MJXp-Span-1468" class="MJXp-mo">−</span><span id="MJXp-Span-1469" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1466" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span><span id="MJXp-Span-1470" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1471" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1472" class="MJXp-mi MJXp-italic MJXp-script">j</span></span><span id="MJXp-Span-1473" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1474" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1475" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1476" class="MJXp-mo">∈</span><span id="MJXp-Span-1477" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-243-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1737" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1738" class="mrow"><span id="MathJax-Span-1739" class="msubsup"><span style="display: inline-block; position: relative; width: 0.924em; height: 0px;"><span style="position: absolute; clip: rect(3.3em, 1000.3em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1740" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span id="MathJax-Span-1741" class="texatom"><span id="MathJax-Span-1742" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1743" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1744" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1745" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1746" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1747" class="msubsup"><span style="display: inline-block; position: relative; width: 1.413em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1748" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.91em, 4.135em, -1000em); top: -4.335em; left: 0.5em;"><span id="MathJax-Span-1749" class="texatom"><span id="MathJax-Span-1750" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1751" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1752" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.385em, 1000.27em, 4.281em, -1000em); top: -3.694em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1753" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1754" class="msubsup"><span style="display: inline-block; position: relative; width: 1.413em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1755" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.91em, 4.135em, -1000em); top: -4.335em; left: 0.5em;"><span id="MathJax-Span-1756" class="texatom"><span id="MathJax-Span-1757" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1758" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1759" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.27em, 4.142em, -1000em); top: -3.692em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1760" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1761" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1762" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1763" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1764" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1765" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1766" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1767" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1768" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-243" type="math/tex">t_{i,j} = u_j^{-1} u_i^{-1} u_j u_i \in R</script></span>.</p>
-
-<p>* If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1478" class="MJXp-math"><span id="MJXp-Span-1479" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1480" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1481" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-244-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1769" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1770" class="mrow"><span id="MathJax-Span-1771" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1772" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1773" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-244" type="math/tex">g_i</script></span> has finite order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1482" class="MJXp-math"><span id="MJXp-Span-1483" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1484" class="MJXp-mi MJXp-italic">d</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1485" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-245-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1774" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1775" class="mrow"><span id="MathJax-Span-1776" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.53em, 4.148em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1777" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1778" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-245" type="math/tex">d_i</script></span>, let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1486" class="MJXp-math"><span id="MJXp-Span-1487" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1488" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1489" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1490" class="MJXp-mo">=</span><span id="MJXp-Span-1491" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1492" class="MJXp-mi MJXp-italic">u</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.59em;"><span id="MJXp-Span-1494" class="MJXp-mrow"><span id="MJXp-Span-1495" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1496" class="MJXp-mi MJXp-italic">d</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1497" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1493" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1498" class="MJXp-mo">∈</span><span id="MJXp-Span-1499" class="MJXp-mi MJXp-italic">R</span></span></span><span style="" tabindex="0" id="MathJax-Element-246-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1779" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1780" class="mrow"><span id="MathJax-Span-1781" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1782" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1783" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1784" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1785" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1786" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.62em, 4.246em, -1000em); top: -4.446em; left: 0.5em;"><span id="MathJax-Span-1787" class="texatom"><span id="MathJax-Span-1788" class="mrow"><span id="MathJax-Span-1789" class="msubsup"><span style="display: inline-block; position: relative; width: 0.546em; height: 0px;"><span style="position: absolute; clip: rect(3.363em, 1000.37em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1790" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1791" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.27em, 4.142em, -1000em); top: -3.692em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1792" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1793" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1794" class="mi">R</span></span></span></span></span></nobr></span><script id="MathJax-Element-246" type="math/tex">b_i=u_i^{d_i} \in R</script></span>.</p>
-
-<p>Then</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1500" class="MJXp-math MJXp-display"><span id="MJXp-Span-1501" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1502" class="MJXp-mo">∗</span><span id="MJXp-Span-1503" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1504" class="MJXp-mo">=</span><span id="MJXp-Span-1505" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1506" class="MJXp-mo">[</span><span id="MJXp-Span-1507" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1508" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1509" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1510" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1511" class="MJXp-mo">…</span><span style="margin-left: 0em; 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clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1861" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1862" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1863" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1864" class="mn">1</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1865" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1866" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1867" class="mo"><</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1868" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1869" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1870" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1871" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1872" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1873" class="mo">,</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-247" type="math/tex; mode=display">
- R*G = R[u_1,\dots,u_n | ru_i = u_i r^{a_i}, u_j u_i = t_{ij} u_i u_j, u_i^{d_i} = b_i (1 \le i < j \le n) ],
- </script></p>
-
-<p>where the last relation vanishes if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1574" class="MJXp-math"><span id="MJXp-Span-1575" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1576" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1577" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-248-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1874" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1875" class="mrow"><span id="MathJax-Span-1876" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1877" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1878" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-248" type="math/tex">g_i</script></span> has infinite order.</p>
-
-<p>Therefore <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1578" class="MJXp-math"><span id="MJXp-Span-1579" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1580" class="MJXp-mo">∗</span><span id="MJXp-Span-1581" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-249-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1879" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1880" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1881" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1882" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1883" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-249" type="math/tex">R*G</script></span> is completely determined by the following data:</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1582" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1583" class="MJXp-mo">[</span><span id="MJXp-Span-1584" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1585" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1586" class="MJXp-mo">[</span><span id="MJXp-Span-1587" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1588" class="MJXp-mi MJXp-italic">d</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1589" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1590" class="MJXp-mo">,</span><span id="MJXp-Span-1591" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1592" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1593" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1594" class="MJXp-mo">,</span><span id="MJXp-Span-1595" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1596" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1597" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span id="MJXp-Span-1598" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-1599" class="MJXp-mo">]</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-1604" class="MJXp-mi MJXp-italic">n</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1600" class="MJXp-mrow"><span id="MJXp-Span-1601" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-1602" class="MJXp-mo">=</span><span id="MJXp-Span-1603" class="MJXp-mn">1</span></span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1605" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1606" class="MJXp-mo">[</span><span id="MJXp-Span-1607" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1608" class="MJXp-mi MJXp-italic">t</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1609" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1610" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-1611" class="MJXp-mo">,</span><span id="MJXp-Span-1612" class="MJXp-mi MJXp-italic">j</span></span></span><span id="MJXp-Span-1613" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-1614" class="MJXp-mo">]</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1615" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1616" class="MJXp-mn">1</span><span id="MJXp-Span-1617" class="MJXp-mo">≤</span><span id="MJXp-Span-1618" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-1619" class="MJXp-mo"><</span><span id="MJXp-Span-1620" class="MJXp-mi MJXp-italic">j</span><span id="MJXp-Span-1621" class="MJXp-mo">≤</span><span id="MJXp-Span-1622" class="MJXp-mi MJXp-italic">n</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1623" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1624" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-250-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-1884" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1885" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1886" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1887" class="mi">R</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1888" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1889" class="mo">[</span><span id="MathJax-Span-1890" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.53em, 4.148em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1891" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1892" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1893" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1894" class="msubsup"><span style="display: inline-block; position: relative; width: 0.773em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1895" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.501em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1896" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1897" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1898" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1899" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1900" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1901" class="msubsup"><span style="display: inline-block; position: relative; width: 1.442em; height: 0px;"><span style="position: absolute; clip: rect(3.184em, 1000.24em, 4.291em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-1902" class="mo">]</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.141em, -1000em); top: -4.341em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1903" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.11em, 4.142em, -1000em); top: -3.677em; left: 0.333em;"><span id="MathJax-Span-1904" class="texatom"><span id="MathJax-Span-1905" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1906" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1907" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1908" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1909" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1910" class="mo">[</span><span id="MathJax-Span-1911" class="msubsup"><span style="display: inline-block; position: relative; width: 0.924em; height: 0px;"><span style="position: absolute; clip: rect(3.3em, 1000.3em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1912" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span id="MathJax-Span-1913" class="texatom"><span id="MathJax-Span-1914" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1915" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1916" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1917" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1918" class="msubsup"><span style="display: inline-block; position: relative; width: 2.962em; height: 0px;"><span style="position: absolute; clip: rect(3.184em, 1000.24em, 4.291em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-1919" class="mo">]</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.333em;"><span id="MathJax-Span-1920" class="texatom"><span id="MathJax-Span-1921" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1922" class="mn">1</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1923" class="mo">≤</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1924" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1925" class="mo"><</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1926" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1927" class="mo">≤</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1928" class="mi">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1929" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1930" class="mo">.</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-250" type="math/tex; mode=display">
- [ R , [ d_i , a_i , b_i ]_{i=1}^n, [ t_{i,j} ]_{1 \le i < j \le n} ].
- </script></p>
-
-<p><a id="X80BABE5078A29793" name="X80BABE5078A29793"></a></p>
-
-<h4>9.9 <span class="Heading">Classical crossed products</span></h4>
-
-<p>A <em>classical crossed product</em> is a crossed product <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1625" class="MJXp-math"><span id="MJXp-Span-1626" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1627" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-1628" class="MJXp-mo">∗</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-1630" class="MJXp-mi MJXp-italic">t</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1629" class="MJXp-mi MJXp-italic">a</span></span></span></span></span></span><span id="MJXp-Span-1631" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-251-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1931" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1932" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1933" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="padding-left: 0.25em;" id="MathJax-Span-1934" class="msubsup"><span style="display: inline-block; position: relative; width: 0.952em; height: 0px;"><span style="position: absolute; clip: rect(3.375em, 1000.46em, 4.102em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-1935" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.46em, 1000.28em, 4.142em, -1000em); top: -4.343em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1936" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.013em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.142em, -1000em); top: -3.843em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1937" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1938" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-251" type="math/tex">L*_a^t G</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1632" class="MJXp-math"><span id="MJXp-Span-1633" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1634" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1635" class="MJXp-mo">/</span></span><span id="MJXp-Span-1636" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-252-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1939" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1940" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1941" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-1942" class="texatom"><span id="MathJax-Span-1943" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1944" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1945" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-252" type="math/tex">L/K</script></span> is a finite Galois extension, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1637" class="MJXp-math"><span id="MJXp-Span-1638" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1639" class="MJXp-mo">=</span><span id="MJXp-Span-1640" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-1641" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-1642" class="MJXp-mi MJXp-italic">l</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1643" class="MJXp-mo">(</span><span id="MJXp-Span-1644" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1645" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1646" class="MJXp-mo">/</span></span><span id="MJXp-Span-1647" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1648" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-253-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1946" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1947" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1948" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1949" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1950" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1951" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1952" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1953" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1954" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-1955" class="texatom"><span id="MathJax-Span-1956" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1957" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1958" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1959" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-253" type="math/tex">G=Gal(L/K)</script></span> is the Galois group of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1649" class="MJXp-math"><span id="MJXp-Span-1650" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1651" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1652" class="MJXp-mo">/</span></span><span id="MJXp-Span-1653" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-254-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1960" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1961" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1962" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-1963" class="texatom"><span id="MathJax-Span-1964" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1965" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1966" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-254" type="math/tex">L/K</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1654" class="MJXp-math"><span id="MJXp-Span-1655" class="MJXp-mi MJXp-italic">a</span></span></span><span style="" tabindex="0" id="MathJax-Element-255-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1967" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1968" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1969" class="mi">a</span></span></span></span></span></nobr></span><script id="MathJax-Element-255" type="math/tex">a</script></span> is the natural action of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1656" class="MJXp-math"><span id="MJXp-Span-1657" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-256-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1970" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1971" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1972" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-256" type="math/tex">G</script></span> on <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1658" class="MJXp-math"><span id="MJXp-Span-1659" class="MJXp-mi MJXp-italic">L</span></span></span><span style="" tabindex="0" id="MathJax-Element-257-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1973" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1974" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1975" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-257" type="math/tex">L</script></span>. Then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1660" class="MJXp-math"><span id="MJXp-Span-1661" class="MJXp-mi MJXp-italic">t</span></span></span><span style="" tabindex="0" id="MathJax-Element-258-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1976" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1977" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1978" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-258" type="math/tex">t</script></span> is a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1662" class="MJXp-math"><span id="MJXp-Span-1663" class="MJXp-mn">2</span></span></span><span style="" tabindex="0" id="MathJax-Element-259-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1979" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1980" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1981" class="mn">2</span></span></span></span></span></nobr></span><script id="MathJax-Element-259" type="math/tex">2</script></span>-cocycle and the <em>crossed product</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7FB21779832CE1CB"><span class="RefLink">9.6</span></a>) <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1664" class="MJXp-math"><span id="MJXp-Span-1665" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1666" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-1667" class="MJXp-mo">∗</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-1669" class="MJXp-mi MJXp-italic">t</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-1668" class="MJXp-mi MJXp-italic">a</span></span></span></span></span></span><span id="MJXp-Span-1670" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-260-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1982" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1983" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1984" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="padding-left: 0.25em;" id="MathJax-Span-1985" class="msubsup"><span style="display: inline-block; position: relative; width: 0.952em; height: 0px;"><span style="position: absolute; clip: rect(3.375em, 1000.46em, 4.102em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-1986" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.46em, 1000.28em, 4.142em, -1000em); top: -4.343em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1987" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.013em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.142em, -1000em); top: -3.843em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1988" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1989" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-260" type="math/tex">L*_a^t G</script></span> is denoted by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1671" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1672" class="MJXp-mo">(</span><span id="MJXp-Span-1673" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1674" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1675" class="MJXp-mo">/</span></span><span id="MJXp-Span-1676" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1677" class="MJXp-mo">,</span><span id="MJXp-Span-1678" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1679" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-261-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-1990" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-1991" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1992" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1993" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-1994" class="texatom"><span id="MathJax-Span-1995" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1996" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1997" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1998" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1999" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2000" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-261" type="math/tex">(L/K,t)</script></span>. The crossed product <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1680" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1681" class="MJXp-mo">(</span><span id="MJXp-Span-1682" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1683" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1684" class="MJXp-mo">/</span></span><span id="MJXp-Span-1685" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1686" class="MJXp-mo">,</span><span id="MJXp-Span-1687" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1688" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-262-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2001" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2002" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2003" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2004" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-2005" class="texatom"><span id="MathJax-Span-2006" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2007" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2008" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2009" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2010" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2011" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-262" type="math/tex">(L/K,t)</script></span> is known to be a central simple <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1689" class="MJXp-math"><span id="MJXp-Span-1690" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-263-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2012" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2013" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2014" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-263" type="math/tex">K</script></span>-algebra <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBR">[Rei03]</a>.</p>
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-<p><a id="X84C98BB8859BBEE2" name="X84C98BB8859BBEE2"></a></p>
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-<h4>9.10 <span class="Heading">Cyclic Algebras</span></h4>
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-<p>A <em>cyclic algebra</em> is a <em>classical crossed product</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X80BABE5078A29793"><span class="RefLink">9.9</span></a>) <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1691" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1692" class="MJXp-mo">(</span><span id="MJXp-Span-1693" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1694" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1695" class="MJXp-mo">/</span></span><span id="MJXp-Span-1696" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1697" class="MJXp-mo">,</span><span id="MJXp-Span-1698" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1699" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-264-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2015" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2016" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2017" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2018" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-2019" class="texatom"><span id="MathJax-Span-2020" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2021" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2022" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2023" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2024" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2025" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-264" type="math/tex">(L/K,t)</script></span> where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1700" class="MJXp-math"><span id="MJXp-Span-1701" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1702" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1703" class="MJXp-mo">/</span></span><span id="MJXp-Span-1704" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-265-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2026" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2027" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2028" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-2029" class="texatom"><span id="MathJax-Span-2030" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2031" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2032" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-265" type="math/tex">L/K</script></span> is a finite cyclic field extension. The cyclic algebras have a very simple form.</p>
-
-<p>Assume that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1705" class="MJXp-math"><span id="MJXp-Span-1706" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-1707" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-1708" class="MJXp-mi MJXp-italic">l</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1709" class="MJXp-mo">(</span><span id="MJXp-Span-1710" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1711" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1712" class="MJXp-mo">/</span></span><span id="MJXp-Span-1713" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1714" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-266-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2033" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2034" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2035" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2036" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2037" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2038" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2039" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-2040" class="texatom"><span id="MathJax-Span-2041" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2042" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2043" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2044" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-266" type="math/tex">Gal(L/K)</script></span> is generated by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1715" class="MJXp-math"><span id="MJXp-Span-1716" class="MJXp-mi MJXp-italic">g</span></span></span><span style="" tabindex="0" id="MathJax-Element-267-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2045" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2046" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2047" class="mi">g</span></span></span></span></span></nobr></span><script id="MathJax-Element-267" type="math/tex">g</script></span> and has order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1717" class="MJXp-math"><span id="MJXp-Span-1718" class="MJXp-mi MJXp-italic">d</span></span></span><span style="" tabindex="0" id="MathJax-Element-268-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2048" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2049" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2050" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-268" type="math/tex">d</script></span>. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1719" class="MJXp-math"><span id="MJXp-Span-1720" class="MJXp-mi MJXp-italic">u</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1721" class="MJXp-mo">=</span><span id="MJXp-Span-1722" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1723" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1724" class="MJXp-mi MJXp-italic MJXp-script">g</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-269-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2051" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2052" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2053" class="mi">u</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2054" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2055" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2056" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2057" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-269" type="math/tex">u=u_g</script></span> be the basis unit (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7FB21779832CE1CB"><span class="RefLink">9.6</span></a>) of the crossed product corresponding to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1725" class="MJXp-math"><span id="MJXp-Span-1726" class="MJXp-mi MJXp-italic">g</span></span></span><span style="" tabindex="0" id="MathJax-Element-270-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2058" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2059" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2060" class="mi">g</span></span></span></span></span></nobr></span><script id="MathJax-Element-270" type="math/tex">g</script></span> and take the remaining basis units for the crossed product by setting <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1727" class="MJXp-math"><span id="MJXp-Span-1728" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1729" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1730" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-1731" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1732" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1733" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1734" class="MJXp-mo">=</span><span id="MJXp-Span-1735" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1736" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1737" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-271-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2061" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2062" class="mrow"><span id="MathJax-Span-2063" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2064" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.804em; left: 0.5em;"><span id="MathJax-Span-2065" class="texatom"><span id="MathJax-Span-2066" class="mrow"><span id="MathJax-Span-2067" class="msubsup"><span style="display: inline-block; position: relative; width: 0.546em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.33em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2068" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.195em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2069" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2070" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2071" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2072" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2073" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-271" type="math/tex">u_{g^i} = u^i</script></span>, (<span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1738" class="MJXp-math"><span id="MJXp-Span-1739" class="MJXp-mi MJXp-italic">i</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1740" class="MJXp-mo">=</span><span id="MJXp-Span-1741" class="MJXp-mn">0</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1742" class="MJXp-mo">,</span><span id="MJXp-Span-1743" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1744" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1745" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1746" class="MJXp-mo">,</span><span id="MJXp-Span-1747" class="MJXp-mi MJXp-italic">d</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1748" class="MJXp-mo">−</span><span id="MJXp-Span-1749" class="MJXp-mn">1</span></span></span><span style="" tabindex="0" id="MathJax-Element-272-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2074" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2075" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2076" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2077" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2078" class="mn">0</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2079" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2080" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2081" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2082" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2083" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2084" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2085" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2086" class="mn">1</span></span></span></span></span></nobr></span><script id="MathJax-Element-272" type="math/tex"> i = 0, 1, \dots, d-1 </script></span>). Then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1750" class="MJXp-math"><span id="MJXp-Span-1751" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1752" class="MJXp-mo">=</span><span id="MJXp-Span-1753" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1754" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1755" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1756" class="MJXp-mo">∈</span><span id="MJXp-Span-1757" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-273-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2087" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2088" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2089" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2090" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2091" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2092" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2093" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2094" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2095" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-273" type="math/tex">a = u^n \in K</script></span>. The cyclic algebra is usually denoted by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1758" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1759" class="MJXp-mo">(</span><span id="MJXp-Span-1760" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1761" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1762" class="MJXp-mo">/</span></span><span id="MJXp-Span-1763" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1764" class="MJXp-mo">,</span><span id="MJXp-Span-1765" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1766" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-274-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2096" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2097" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2098" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2099" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-2100" class="texatom"><span id="MathJax-Span-2101" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2102" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2103" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2104" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2105" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2106" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-274" type="math/tex">(L/K,a)</script></span> and one has the following description of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1767" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1768" class="MJXp-mo">(</span><span id="MJXp-Span-1769" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1770" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1771" class="MJXp-mo">/</span></span><span id="MJXp-Span-1772" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1773" class="MJXp-mo">,</span><span id="MJXp-Span-1774" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1775" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-275-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2107" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2108" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2109" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2110" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-2111" class="texatom"><span id="MathJax-Span-2112" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2113" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2114" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2115" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2116" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2117" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-275" type="math/tex">(L/K,t)</script></span></p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1776" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1777" class="MJXp-mo">(</span><span id="MJXp-Span-1778" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1779" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1780" class="MJXp-mo">/</span></span><span id="MJXp-Span-1781" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1782" class="MJXp-mo">,</span><span id="MJXp-Span-1783" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1784" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1785" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1786" class="MJXp-mo">(</span><span id="MJXp-Span-1787" class="MJXp-mi MJXp-italic">L</span><span id="MJXp-Span-1788" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1789" class="MJXp-mo">/</span></span><span id="MJXp-Span-1790" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1791" class="MJXp-mo">,</span><span id="MJXp-Span-1792" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1793" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1794" class="MJXp-mo">=</span><span id="MJXp-Span-1795" class="MJXp-mi MJXp-italic">L</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1796" class="MJXp-mo">[</span><span id="MJXp-Span-1797" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-1798" class="MJXp-mrow"><span style="margin-left: 0.167em; margin-right: 0.167em;" id="MJXp-Span-1799" class="MJXp-mo">|</span></span><span id="MJXp-Span-1800" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-1801" class="MJXp-mi MJXp-italic">u</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1802" class="MJXp-mo">=</span><span id="MJXp-Span-1803" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-1804" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1805" class="MJXp-mi MJXp-italic">r</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1806" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1807" class="MJXp-mo">,</span><span id="MJXp-Span-1808" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1809" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1810" class="MJXp-mi MJXp-italic MJXp-script">d</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1811" class="MJXp-mo">=</span><span id="MJXp-Span-1812" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1813" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1814" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-276-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-2118" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2119" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2120" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2121" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-2122" class="texatom"><span id="MathJax-Span-2123" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2124" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2125" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2126" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2127" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2128" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2129" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2130" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2131" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-2132" class="texatom"><span id="MathJax-Span-2133" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2134" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2135" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2136" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2137" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2138" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2139" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2140" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2141" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2142" class="mi">u</span><span id="MathJax-Span-2143" class="texatom"><span id="MathJax-Span-2144" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2145" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2146" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2147" class="mi">u</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2148" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2149" class="mi">u</span><span id="MathJax-Span-2150" class="msubsup"><span style="display: inline-block; position: relative; width: 0.897em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2151" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.469em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2152" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2153" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-2154" class="msubsup"><span style="display: inline-block; position: relative; width: 0.948em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2155" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2156" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2157" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2158" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2159" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2160" class="mo">.</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-276" type="math/tex; mode=display">
- (L/K,t) = (L/K,a) = L[u| r u = u r^g, u^d = a ].
- </script></p>
-
-<p><a id="X8099A8C784255672" name="X8099A8C784255672"></a></p>
-
-<h4>9.11 <span class="Heading">Cyclotomic algebras</span></h4>
-
-<p>A <em>cyclotomic algebra</em> over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1815" class="MJXp-math"><span id="MJXp-Span-1816" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-277-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2161" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2162" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2163" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-277" type="math/tex">F</script></span> is a <em>classical crossed product</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X80BABE5078A29793"><span class="RefLink">9.9</span></a>) <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1817" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1818" class="MJXp-mo">(</span><span id="MJXp-Span-1819" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1820" class="MJXp-mo">(</span><span id="MJXp-Span-1821" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1822" class="MJXp-mo">)</span><span id="MJXp-Span-1823" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1824" class="MJXp-mo">/</span></span><span id="MJXp-Span-1825" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1826" class="MJXp-mo">,</span><span id="MJXp-Span-1827" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1828" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-278-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2164" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2165" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2166" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2167" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2168" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2169" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2170" class="mo">)</span><span id="MathJax-Span-2171" class="texatom"><span id="MathJax-Span-2172" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2173" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2174" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2175" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2176" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2177" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-278" type="math/tex">(F(\xi)/F,t)</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1829" class="MJXp-math"><span id="MJXp-Span-1830" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-279-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2178" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2179" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2180" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-279" type="math/tex">F</script></span> is a field, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1831" class="MJXp-math"><span id="MJXp-Span-1832" class="MJXp-mi MJXp-italic">ξ</span></span></span><span style="" tabindex="0" id="MathJax-Element-280-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2181" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2182" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2183" class="mi">ξ</span></span></span></span></span></nobr></span><script id="MathJax-Element-280" type="math/tex">\xi</script></span> is a root of unity in an extension of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1833" class="MJXp-math"><span id="MJXp-Span-1834" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-281-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2184" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2185" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2186" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-281" type="math/tex">F</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1835" class="MJXp-math"><span id="MJXp-Span-1836" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1837" class="MJXp-mo">(</span><span id="MJXp-Span-1838" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1839" class="MJXp-mo">,</span><span id="MJXp-Span-1840" class="MJXp-mi MJXp-italic">h</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1841" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-282-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2187" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2188" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2189" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2190" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2191" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2192" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2193" class="mi">h</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2194" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-282" type="math/tex">t(g,h)</script></span> is a root of unity for every <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1842" class="MJXp-math"><span id="MJXp-Span-1843" class="MJXp-mi MJXp-italic">g</span></span></span><span style="" tabindex="0" id="MathJax-Element-283-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2195" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2196" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2197" class="mi">g</span></span></span></span></span></nobr></span><script id="MathJax-Element-283" type="math/tex">g</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1844" class="MJXp-math"><span id="MJXp-Span-1845" class="MJXp-mi MJXp-italic">h</span></span></span><span style="" tabindex="0" id="MathJax-Element-284-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2198" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2199" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2200" class="mi">h</span></span></span></span></span></nobr></span><script id="MathJax-Element-284" type="math/tex">h</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1846" class="MJXp-math"><span id="MJXp-Span-1847" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-1848" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-1849" class="MJXp-mi MJXp-italic">l</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1850" class="MJXp-mo">(</span><span id="MJXp-Span-1851" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1852" class="MJXp-mo">(</span><span id="MJXp-Span-1853" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1854" class="MJXp-mo">)</span><span id="MJXp-Span-1855" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1856" class="MJXp-mo">/</span></span><span id="MJXp-Span-1857" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1858" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-285-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2201" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2202" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2203" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2204" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2205" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2206" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2207" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2208" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2209" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2210" class="mo">)</span><span id="MathJax-Span-2211" class="texatom"><span id="MathJax-Span-2212" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2213" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2214" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2215" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-285" type="math/tex">Gal(F(\xi)/F)</script></span>.</p>
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-<p>The <em>Brauer-Witt Theorem</em> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBY">[Yam74]</a> asserts that every <em>Wedderburn component</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of a group algebra is <em>Brauer equivalent</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) (over its centre) to a cyclotomic algebra.</p>
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-<p><a id="X84A142407B7565E0" name="X84A142407B7565E0"></a></p>
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-<h4>9.12 <span class="Heading">Numerical description of cyclotomic algebras</span></h4>
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-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1859" class="MJXp-math"><span id="MJXp-Span-1860" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1861" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1862" class="MJXp-mo">(</span><span id="MJXp-Span-1863" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1864" class="MJXp-mo">(</span><span id="MJXp-Span-1865" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1866" class="MJXp-mo">)</span><span id="MJXp-Span-1867" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1868" class="MJXp-mo">/</span></span><span id="MJXp-Span-1869" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1870" class="MJXp-mo">,</span><span id="MJXp-Span-1871" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1872" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-286-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2216" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2217" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2218" class="mi">A</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2219" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2220" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2221" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2222" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2223" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2224" class="mo">)</span><span id="MathJax-Span-2225" class="texatom"><span id="MathJax-Span-2226" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2227" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2228" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2229" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2230" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2231" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-286" type="math/tex">A=(F(\xi)/F,t)</script></span> be a <em>cyclotomic algebra</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>), where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1873" class="MJXp-math"><span id="MJXp-Span-1874" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1875" class="MJXp-mo">=</span><span id="MJXp-Span-1876" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1877" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1878" class="MJXp-mi MJXp-italic MJXp-script">k</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-287-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2232" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2233" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2234" class="mi">ξ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2235" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2236" class="msubsup"><span style="display: inline-block; position: relative; width: 0.855em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2237" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.454em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2238" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-287" type="math/tex">\xi=\xi_k</script></span> is a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1879" class="MJXp-math"><span id="MJXp-Span-1880" class="MJXp-mi MJXp-italic">k</span></span></span><span style="" tabindex="0" id="MathJax-Element-288-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2239" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2240" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2241" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-288" type="math/tex">k</script></span>-th root of unity. Then the Galois group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1881" class="MJXp-math"><span id="MJXp-Span-1882" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1883" class="MJXp-mo">=</span><span id="MJXp-Span-1884" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-1885" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-1886" class="MJXp-mi MJXp-italic">l</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1887" class="MJXp-mo">(</span><span id="MJXp-Span-1888" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1889" class="MJXp-mo">(</span><span id="MJXp-Span-1890" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1891" class="MJXp-mo">)</span><span id="MJXp-Span-1892" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-1893" class="MJXp-mo">/</span></span><span id="MJXp-Span-1894" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1895" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-289-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2242" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2243" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2244" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2245" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2246" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2247" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2248" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2249" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2250" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2251" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2252" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2253" class="mo">)</span><span id="MathJax-Span-2254" class="texatom"><span id="MathJax-Span-2255" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2256" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2257" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2258" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-289" type="math/tex">G=Gal(F(\xi)/F)</script></span> is abelian and therefore one can obtain a simplified form for the description of cyclotomic algebras as for any <em>abelian crossed product</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7869E2A48784C232"><span class="RefLink">9.8</span></a>).</p>
-
-<p>Then the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1896" class="MJXp-math"><span id="MJXp-Span-1897" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-1898" class="MJXp-mo">×</span><span id="MJXp-Span-1899" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-290-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2259" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2260" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2261" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2262" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2263" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-290" type="math/tex">n \times n</script></span> matrix algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1900" class="MJXp-math"><span id="MJXp-Span-1901" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1902" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1903" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1904" class="MJXp-mo">(</span><span id="MJXp-Span-1905" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1906" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-291-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2264" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2265" class="mrow"><span id="MathJax-Span-2266" class="msubsup"><span style="display: inline-block; position: relative; width: 1.262em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2267" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2268" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2269" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2270" class="mi">A</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2271" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-291" type="math/tex">M_n(A)</script></span> can be described numerically in one of the following forms:</p>
-
-<p>* If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1907" class="MJXp-math"><span id="MJXp-Span-1908" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1909" class="MJXp-mo">(</span><span id="MJXp-Span-1910" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1911" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1912" class="MJXp-mo">=</span><span id="MJXp-Span-1913" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-292-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2272" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2273" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2274" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2275" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2276" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2277" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2278" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2279" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-292" type="math/tex">F(\xi)=F</script></span>, (i.e. <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1914" class="MJXp-math"><span id="MJXp-Span-1915" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1916" class="MJXp-mo">=</span><span id="MJXp-Span-1917" class="MJXp-mn">1</span></span></span><span style="" tabindex="0" id="MathJax-Element-293-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2280" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2281" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2282" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2283" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2284" class="mn">1</span></span></span></span></span></nobr></span><script id="MathJax-Element-293" type="math/tex">G=1</script></span>) then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1918" class="MJXp-math"><span id="MJXp-Span-1919" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1920" class="MJXp-mo">=</span><span id="MJXp-Span-1921" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1922" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1923" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1924" class="MJXp-mo">(</span><span id="MJXp-Span-1925" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1926" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-294-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2285" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2286" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2287" class="mi">A</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2288" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2289" class="msubsup"><span style="display: inline-block; position: relative; width: 1.262em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2290" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2291" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2292" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2293" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2294" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-294" type="math/tex">A=M_n(F)</script></span> and thus the only data needed to describe <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1927" class="MJXp-math"><span id="MJXp-Span-1928" class="MJXp-mi MJXp-italic">A</span></span></span><span style="" tabindex="0" id="MathJax-Element-295-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2295" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2296" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2297" class="mi">A</span></span></span></span></span></nobr></span><script id="MathJax-Element-295" type="math/tex">A</script></span> are the matrix size <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1929" class="MJXp-math"><span id="MJXp-Span-1930" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-296-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2298" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2299" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2300" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-296" type="math/tex">n</script></span> and the field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1931" class="MJXp-math"><span id="MJXp-Span-1932" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-297-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2301" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2302" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2303" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-297" type="math/tex">F</script></span>:</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1933" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1934" class="MJXp-mo">[</span><span id="MJXp-Span-1935" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1936" class="MJXp-mo">,</span><span id="MJXp-Span-1937" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1938" class="MJXp-mo">]</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-298-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-2304" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2305" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2306" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2307" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2308" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2309" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2310" class="mo">]</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-298" type="math/tex; mode=display">
- [n,F]
- </script></p>
-
-<p>* If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1939" class="MJXp-math"><span id="MJXp-Span-1940" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-299-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2311" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2312" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2313" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-299" type="math/tex">G</script></span> is cyclic (but not trivial) of order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1941" class="MJXp-math"><span id="MJXp-Span-1942" class="MJXp-mi MJXp-italic">d</span></span></span><span style="" tabindex="0" id="MathJax-Element-300-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2314" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2315" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2316" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-300" type="math/tex">d</script></span> then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1943" class="MJXp-math"><span id="MJXp-Span-1944" class="MJXp-mi MJXp-italic">A</span></span></span><span style="" tabindex="0" id="MathJax-Element-301-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2317" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2318" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2319" class="mi">A</span></span></span></span></span></nobr></span><script id="MathJax-Element-301" type="math/tex">A</script></span> is a cyclic cyclotomic algebra</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1945" class="MJXp-math MJXp-display"><span id="MJXp-Span-1946" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1947" class="MJXp-mo">=</span><span id="MJXp-Span-1948" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1949" class="MJXp-mo">(</span><span id="MJXp-Span-1950" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1951" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1952" class="MJXp-mo">[</span><span id="MJXp-Span-1953" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-1954" class="MJXp-mrow"><span style="margin-left: 0.167em; margin-right: 0.167em;" id="MJXp-Span-1955" class="MJXp-mo">|</span></span><span id="MJXp-Span-1956" class="MJXp-mi MJXp-italic">ξ</span><span id="MJXp-Span-1957" class="MJXp-mi MJXp-italic">u</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1958" class="MJXp-mo">=</span><span id="MJXp-Span-1959" class="MJXp-mi MJXp-italic">u</span><span id="MJXp-Span-1960" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1961" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1962" class="MJXp-mi MJXp-italic MJXp-script">α</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1963" class="MJXp-mo">,</span><span id="MJXp-Span-1964" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1965" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1966" class="MJXp-mi MJXp-italic MJXp-script">d</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-1967" class="MJXp-mo">=</span><span id="MJXp-Span-1968" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1969" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-1970" class="MJXp-mi MJXp-italic MJXp-script">β</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1971" class="MJXp-mo">]</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-302-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-2320" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2321" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2322" class="mi">A</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2323" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2324" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2325" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2326" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2327" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2328" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2329" class="mi">u</span><span id="MathJax-Span-2330" class="texatom"><span id="MathJax-Span-2331" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2332" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2333" class="mi">ξ</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2334" class="mi">u</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2335" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2336" class="mi">u</span><span id="MathJax-Span-2337" class="msubsup"><span style="display: inline-block; position: relative; width: 0.919em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2338" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2339" class="mi">α</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2340" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-2341" class="msubsup"><span style="display: inline-block; position: relative; width: 0.948em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2342" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2343" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2344" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2345" class="msubsup"><span style="display: inline-block; position: relative; width: 0.892em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2346" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2347" class="mi">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.006em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2348" class="mo">]</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-302" type="math/tex; mode=display">
- A = F(\xi) [ u | \xi u = u \xi^\alpha, u^d = \xi^\beta ]
- </script></p>
-
-<p>and so <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1972" class="MJXp-math"><span id="MJXp-Span-1973" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-1974" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-1975" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1976" class="MJXp-mo">(</span><span id="MJXp-Span-1977" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1978" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-303-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2349" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2350" class="mrow"><span id="MathJax-Span-2351" class="msubsup"><span style="display: inline-block; position: relative; width: 1.262em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2352" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2353" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2354" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2355" class="mi">A</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2356" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-303" type="math/tex">M_n(A)</script></span> can be described with the following data</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1979" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1980" class="MJXp-mo">[</span><span id="MJXp-Span-1981" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1982" class="MJXp-mo">,</span><span id="MJXp-Span-1983" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1984" class="MJXp-mo">,</span><span id="MJXp-Span-1985" class="MJXp-mi MJXp-italic">k</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1986" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1987" class="MJXp-mo">[</span><span id="MJXp-Span-1988" class="MJXp-mi MJXp-italic">d</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1989" class="MJXp-mo">,</span><span id="MJXp-Span-1990" class="MJXp-mi MJXp-italic">α</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1991" class="MJXp-mo">,</span><span id="MJXp-Span-1992" class="MJXp-mi MJXp-italic">β</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1993" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-1994" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-1995" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-304-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-2357" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2358" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2359" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2360" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2361" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2362" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2363" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2364" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2365" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2366" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2367" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2368" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2369" class="mi">α</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2370" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2371" class="mi">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2372" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2373" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2374" class="mo">,</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-304" type="math/tex; mode=display">
- [n,F,k,[d,\alpha,\beta]],
- </script></p>
-
-<p>where the integers <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1996" class="MJXp-math"><span id="MJXp-Span-1997" class="MJXp-mi MJXp-italic">k</span></span></span><span style="" tabindex="0" id="MathJax-Element-305-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2375" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2376" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2377" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-305" type="math/tex">k</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-1998" class="MJXp-math"><span id="MJXp-Span-1999" class="MJXp-mi MJXp-italic">d</span></span></span><span style="" tabindex="0" id="MathJax-Element-306-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2378" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2379" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2380" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-306" type="math/tex">d</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2000" class="MJXp-math"><span id="MJXp-Span-2001" class="MJXp-mi MJXp-italic">α</span></span></span><span style="" tabindex="0" id="MathJax-Element-307-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2381" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2382" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2383" class="mi">α</span></span></span></span></span></nobr></span><script id="MathJax-Element-307" type="math/tex">\alpha</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2002" class="MJXp-math"><span id="MJXp-Span-2003" class="MJXp-mi MJXp-italic">β</span></span></span><span style="" tabindex="0" id="MathJax-Element-308-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2384" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2385" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2386" class="mi">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-308" type="math/tex">\beta</script></span> satisfy the following conditions:</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2004" class="MJXp-math MJXp-display"><span id="MJXp-Span-2005" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2006" class="MJXp-mi MJXp-italic">α</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2007" class="MJXp-mi MJXp-italic MJXp-script">d</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2008" class="MJXp-mo">≡</span><span id="MJXp-Span-2009" class="MJXp-mn">1</span><span style="width: 0.278em; height: 0em;" id="MJXp-Span-2010" class="MJXp-mspace"></span><span id="MJXp-Span-2011" class="MJXp-mi MJXp-italic">m</span><span id="MJXp-Span-2012" class="MJXp-mi MJXp-italic">o</span><span id="MJXp-Span-2013" class="MJXp-mi MJXp-italic">d</span><span style="width: 0.278em; height: 0em;" id="MJXp-Span-2014" class="MJXp-mspace"></span><span id="MJXp-Span-2015" class="MJXp-mi MJXp-italic">k</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2016" class="MJXp-mo">,</span><span style="width: 1em; height: 0em;" id="MJXp-Span-2017" class="MJXp-mspace"></span><span id="MJXp-Span-2018" class="MJXp-mi MJXp-italic">β</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2019" class="MJXp-mo">(</span><span id="MJXp-Span-2020" class="MJXp-mi MJXp-italic">α</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-2021" class="MJXp-mo">−</span><span id="MJXp-Span-2022" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2023" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2024" class="MJXp-mo">≡</span><span id="MJXp-Span-2025" class="MJXp-mn">0</span><span style="width: 0.278em; height: 0em;" id="MJXp-Span-2026" class="MJXp-mspace"></span><span id="MJXp-Span-2027" class="MJXp-mi MJXp-italic">m</span><span id="MJXp-Span-2028" class="MJXp-mi MJXp-italic">o</span><span id="MJXp-Span-2029" class="MJXp-mi MJXp-italic">d</span><span style="width: 0.278em; height: 0em;" id="MJXp-Span-2030" class="MJXp-mspace"></span><span id="MJXp-Span-2031" class="MJXp-mi MJXp-italic">k</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2032" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-309-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-2387" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2388" class="mrow"><span id="MathJax-Span-2389" class="msubsup"><span style="display: inline-block; position: relative; width: 1em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.55em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2390" class="mi">α</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.552em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2391" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2392" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2393" class="mn">1</span><span style="height: 0em; vertical-align: 0em; width: 0.313em; display: inline-block; overflow: hidden;" id="MathJax-Span-2394" class="mspace"></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2395" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2396" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2397" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="height: 0em; vertical-align: 0em; width: 0.313em; display: inline-block; overflow: hidden;" id="MathJax-Span-2398" class="mspace"></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2399" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2400" class="mo">,</span><span style="height: 0em; vertical-align: 0em; width: 1.125em; display: inline-block; overflow: hidden;" id="MathJax-Span-2401" class="mspace"></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2402" class="mi">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2403" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2404" class="mi">α</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2405" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2406" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2407" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2408" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2409" class="mn">0</span><span style="height: 0em; vertical-align: 0em; width: 0.313em; display: inline-block; overflow: hidden;" id="MathJax-Span-2410" class="mspace"></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2411" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2412" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2413" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="height: 0em; vertical-align: 0em; width: 0.313em; display: inline-block; overflow: hidden;" id="MathJax-Span-2414" class="mspace"></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2415" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2416" class="mo">.</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-309" type="math/tex; mode=display">
- \alpha^d \equiv 1 \; mod \; k, \quad
- \beta(\alpha-1) \equiv 0 \; mod \; k.
- </script></p>
-
-<p>* If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2033" class="MJXp-math"><span id="MJXp-Span-2034" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-310-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2417" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2418" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2419" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-310" type="math/tex">G</script></span> is abelian but not cyclic then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2035" class="MJXp-math"><span id="MJXp-Span-2036" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2037" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2038" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2039" class="MJXp-mo">(</span><span id="MJXp-Span-2040" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2041" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-311-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2420" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2421" class="mrow"><span id="MathJax-Span-2422" class="msubsup"><span style="display: inline-block; position: relative; width: 1.262em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2423" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2424" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2425" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2426" class="mi">A</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2427" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-311" type="math/tex">M_n(A)</script></span> can be described with the following data (see <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7869E2A48784C232"><span class="RefLink">9.8</span></a>):</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2042" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2043" class="MJXp-mo">[</span><span id="MJXp-Span-2044" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2045" class="MJXp-mo">,</span><span id="MJXp-Span-2046" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2047" class="MJXp-mo">,</span><span id="MJXp-Span-2048" class="MJXp-mi MJXp-italic">k</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2049" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2050" class="MJXp-mo">[</span><span id="MJXp-Span-2051" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2052" class="MJXp-mi MJXp-italic">d</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2053" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2054" class="MJXp-mo">,</span><span id="MJXp-Span-2055" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2056" class="MJXp-mi MJXp-italic">α</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2057" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2058" class="MJXp-mo">,</span><span id="MJXp-Span-2059" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2060" class="MJXp-mi MJXp-italic">β</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2061" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span id="MJXp-Span-2062" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-2063" class="MJXp-mo">]</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-2068" class="MJXp-mi MJXp-italic">m</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-2064" class="MJXp-mrow"><span id="MJXp-Span-2065" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-2066" class="MJXp-mo">=</span><span id="MJXp-Span-2067" class="MJXp-mn">1</span></span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2069" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2070" class="MJXp-mo">[</span><span id="MJXp-Span-2071" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2072" class="MJXp-mi MJXp-italic">γ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2073" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2074" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-2075" class="MJXp-mo">,</span><span id="MJXp-Span-2076" class="MJXp-mi MJXp-italic">j</span></span></span><span id="MJXp-Span-2077" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-2078" class="MJXp-mo">]</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2079" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2080" class="MJXp-mn">1</span><span id="MJXp-Span-2081" class="MJXp-mo">≤</span><span id="MJXp-Span-2082" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-2083" class="MJXp-mo"><</span><span id="MJXp-Span-2084" class="MJXp-mi MJXp-italic">j</span><span id="MJXp-Span-2085" class="MJXp-mo">≤</span><span id="MJXp-Span-2086" class="MJXp-mi MJXp-italic">m</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2087" class="MJXp-mo">]</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-312-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-2428" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2429" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2430" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2431" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2432" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2433" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2434" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2435" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2436" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2437" class="mo">[</span><span id="MathJax-Span-2438" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.53em, 4.148em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2439" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2440" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2441" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-2442" class="msubsup"><span style="display: inline-block; position: relative; width: 0.824em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.55em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2443" class="mi">α</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.552em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2444" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2445" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-2446" class="msubsup"><span style="display: inline-block; position: relative; width: 0.778em; height: 0px;"><span style="position: absolute; clip: rect(3.168em, 1000.51em, 4.34em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2447" class="mi">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.506em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2448" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-2449" class="msubsup"><span style="display: inline-block; position: relative; width: 1.442em; height: 0px;"><span style="position: absolute; clip: rect(3.184em, 1000.24em, 4.291em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-2450" class="mo">]</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.59em, 4.141em, -1000em); top: -4.341em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2451" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.11em, 4.142em, -1000em); top: -3.677em; left: 0.333em;"><span id="MathJax-Span-2452" class="texatom"><span id="MathJax-Span-2453" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2454" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2455" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2456" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2457" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2458" class="mo">[</span><span id="MathJax-Span-2459" class="msubsup"><span style="display: inline-block; position: relative; width: 1.056em; height: 0px;"><span style="position: absolute; clip: rect(3.411em, 1000.44em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2460" class="mi">γ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.028em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.41em;"><span id="MathJax-Span-2461" class="texatom"><span id="MathJax-Span-2462" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2463" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2464" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2465" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-2466" class="msubsup"><span style="display: inline-block; position: relative; width: 3.119em; height: 0px;"><span style="position: absolute; clip: rect(3.184em, 1000.24em, 4.291em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-2467" class="mo">]</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.333em;"><span id="MathJax-Span-2468" class="texatom"><span id="MathJax-Span-2469" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2470" class="mn">1</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2471" class="mo">≤</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2472" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2473" class="mo"><</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2474" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2475" class="mo">≤</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2476" class="mi">m</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2477" class="mo">]</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-312" type="math/tex; mode=display">
- [n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le m} ]
- </script></p>
-
-<p>representing the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2088" class="MJXp-math"><span id="MJXp-Span-2089" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-2090" class="MJXp-mo">×</span><span id="MJXp-Span-2091" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-313-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2478" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2479" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2480" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2481" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2482" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-313" type="math/tex">n \times n</script></span> matrix ring over the following algebra:</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2092" class="MJXp-math MJXp-display"><span id="MJXp-Span-2093" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2094" class="MJXp-mo">=</span><span id="MJXp-Span-2095" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2096" class="MJXp-mo">(</span><span id="MJXp-Span-2097" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2098" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2099" class="MJXp-mo">[</span><span id="MJXp-Span-2100" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2101" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2102" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2103" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2104" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2105" class="MJXp-mo">,</span><span id="MJXp-Span-2106" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2107" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2108" class="MJXp-mi MJXp-italic MJXp-script">m</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2109" class="MJXp-mo">∣</span><span id="MJXp-Span-2110" class="MJXp-mi MJXp-italic">ξ</span><span id="MJXp-Span-2111" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2112" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2113" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2114" class="MJXp-mo">=</span><span id="MJXp-Span-2115" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2116" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2117" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span id="MJXp-Span-2118" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2119" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2120" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2121" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2122" class="MJXp-mi MJXp-italic">α</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2123" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2124" class="MJXp-mo">,</span><span style="width: 1em; height: 0em;" id="MJXp-Span-2125" class="MJXp-mspace"></span><span id="MJXp-Span-2126" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2127" class="MJXp-mi MJXp-italic">u</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.59em;"><span id="MJXp-Span-2129" class="MJXp-mrow"><span id="MJXp-Span-2130" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2131" class="MJXp-mi MJXp-italic">d</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2132" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-2128" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2133" class="MJXp-mo">=</span><span id="MJXp-Span-2134" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2135" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2136" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2137" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2138" class="MJXp-mi MJXp-italic">β</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2139" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2140" class="MJXp-mo">,</span><span style="width: 1em; height: 0em;" id="MJXp-Span-2141" class="MJXp-mspace"></span><span id="MJXp-Span-2142" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2143" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2144" class="MJXp-mi MJXp-italic MJXp-script">s</span></span><span id="MJXp-Span-2145" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2146" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2147" class="MJXp-mi MJXp-italic MJXp-script">r</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2148" class="MJXp-mo">=</span><span id="MJXp-Span-2149" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2150" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2151" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2152" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2153" class="MJXp-mi MJXp-italic">γ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2154" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2155" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-2156" class="MJXp-mi MJXp-italic">s</span></span></span></span></span><span id="MJXp-Span-2157" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2158" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2159" class="MJXp-mi MJXp-italic MJXp-script">r</span></span><span id="MJXp-Span-2160" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2161" class="MJXp-mi MJXp-italic">u</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2162" class="MJXp-mi MJXp-italic MJXp-script">s</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2163" class="MJXp-mo">,</span><span style="width: 1em; height: 0em;" id="MJXp-Span-2164" class="MJXp-mspace"></span><span id="MJXp-Span-2165" class="MJXp-mi MJXp-italic">i</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2166" class="MJXp-mo">=</span><span id="MJXp-Span-2167" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2168" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2169" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2170" class="MJXp-mo">,</span><span id="MJXp-Span-2171" class="MJXp-mi MJXp-italic">m</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2172" class="MJXp-mo">,</span><span style="width: 1em; height: 0em;" id="MJXp-Span-2173" class="MJXp-mspace"></span><span id="MJXp-Span-2174" class="MJXp-mn">0</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2175" class="MJXp-mo">≤</span><span id="MJXp-Span-2176" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2177" class="MJXp-mo"><</span><span id="MJXp-Span-2178" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2179" class="MJXp-mo">≤</span><span id="MJXp-Span-2180" class="MJXp-mi MJXp-italic">m</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2181" class="MJXp-mo">]</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-314-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-2483" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2484" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2485" class="mi">A</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2486" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2487" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2488" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2489" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2490" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2491" class="mo">[</span><span id="MathJax-Span-2492" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2493" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2494" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2495" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2496" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2497" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-2498" class="msubsup"><span style="display: inline-block; position: relative; width: 1.086em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2499" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2500" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2501" class="mo">∣</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2502" class="mi">ξ</span><span id="MathJax-Span-2503" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2504" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2505" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2506" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2507" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2508" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2509" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-2510" class="msubsup"><span style="display: inline-block; position: relative; width: 1.111em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2511" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span id="MathJax-Span-2512" class="texatom"><span id="MathJax-Span-2513" class="mrow"><span id="MathJax-Span-2514" class="msubsup"><span style="display: inline-block; position: relative; width: 0.582em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.39em, 4.142em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2515" class="mi">α</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.39em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2516" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2517" class="mo">,</span><span style="height: 0em; vertical-align: 0em; width: 1.125em; display: inline-block; overflow: hidden;" id="MathJax-Span-2518" class="mspace"></span><span style="padding-left: 0.188em;" id="MathJax-Span-2519" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2520" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.62em, 4.246em, -1000em); top: -4.446em; left: 0.5em;"><span id="MathJax-Span-2521" class="texatom"><span id="MathJax-Span-2522" class="mrow"><span id="MathJax-Span-2523" class="msubsup"><span style="display: inline-block; position: relative; width: 0.546em; height: 0px;"><span style="position: absolute; clip: rect(3.363em, 1000.37em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2524" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2525" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.27em, 4.142em, -1000em); top: -3.692em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2526" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2527" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2528" class="msubsup"><span style="display: inline-block; position: relative; width: 1.079em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2529" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span id="MathJax-Span-2530" class="texatom"><span id="MathJax-Span-2531" class="mrow"><span id="MathJax-Span-2532" class="msubsup"><span style="display: inline-block; position: relative; width: 0.55em; height: 0px;"><span style="position: absolute; clip: rect(3.367em, 1000.36em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2533" class="mi">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.006em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.82em; left: 0.358em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2534" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2535" class="mo">,</span><span style="height: 0em; vertical-align: 0em; width: 1.125em; display: inline-block; overflow: hidden;" id="MathJax-Span-2536" class="mspace"></span><span style="padding-left: 0.188em;" id="MathJax-Span-2537" class="msubsup"><span style="display: inline-block; position: relative; width: 0.85em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2538" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2539" class="mi">s</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-2540" class="msubsup"><span style="display: inline-block; position: relative; width: 0.866em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2541" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2542" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2543" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2544" class="msubsup"><span style="display: inline-block; position: relative; width: 1.272em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2545" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span id="MathJax-Span-2546" class="texatom"><span id="MathJax-Span-2547" class="mrow"><span id="MathJax-Span-2548" class="msubsup"><span style="display: inline-block; position: relative; width: 0.743em; height: 0px;"><span style="position: absolute; clip: rect(3.539em, 1000.31em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2549" class="mi">γ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.02em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.82em; left: 0.29em;"><span id="MathJax-Span-2550" class="texatom"><span id="MathJax-Span-2551" class="mrow"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2552" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.011em;"></span></span><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2553" class="mi">s</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-2554" class="msubsup"><span style="display: inline-block; position: relative; width: 0.866em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2555" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2556" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-2557" class="msubsup"><span style="display: inline-block; position: relative; width: 0.85em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2558" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2559" class="mi">s</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2560" class="mo">,</span><span style="height: 0em; vertical-align: 0em; width: 1.125em; display: inline-block; overflow: hidden;" id="MathJax-Span-2561" class="mspace"></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2562" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2563" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2564" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2565" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2566" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2567" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2568" class="mi">m</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2569" class="mo">,</span><span style="height: 0em; vertical-align: 0em; width: 1.125em; display: inline-block; overflow: hidden;" id="MathJax-Span-2570" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2571" class="mn">0</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2572" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2573" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2574" class="mo"><</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2575" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2576" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2577" class="mi">m</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2578" class="mo">]</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-314" type="math/tex; mode=display">
- A = F(\xi)[ u_1, \ldots, u_m \mid
- \xi u_i = u_i \xi^{\alpha_i}, \quad
- u_i^{d_i}=\xi^{\beta_i}, \quad
- u_s u_r = \xi^{\gamma_{rs}} u_r u_s, \quad
- i = 1, \ldots, m, \quad
- 0 \le r < s \le m ]
- </script></p>
-
-<p>where</p>
-
-<p>* <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2182" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2183" class="MJXp-mo">{</span><span id="MJXp-Span-2184" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2185" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2186" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2187" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2188" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2189" class="MJXp-mo">,</span><span id="MJXp-Span-2190" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2191" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2192" class="MJXp-mi MJXp-italic MJXp-script">m</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2193" class="MJXp-mo">}</span></span></span><span style="" tabindex="0" id="MathJax-Element-315-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2579" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2580" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2581" class="mo">{</span><span id="MathJax-Span-2582" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2583" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2584" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2585" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2586" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-2587" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-2588" class="msubsup"><span style="display: inline-block; position: relative; width: 1.086em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2589" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2590" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2591" class="mo">}</span></span></span></span></span></nobr></span><script id="MathJax-Element-315" type="math/tex">\{g_1,\ldots,g_m\}</script></span> is an independent set of generators of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2194" class="MJXp-math"><span id="MJXp-Span-2195" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-316-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2592" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2593" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2594" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-316" type="math/tex">G</script></span>,</p>
-
-<p>* <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2196" class="MJXp-math"><span id="MJXp-Span-2197" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2198" class="MJXp-mi MJXp-italic">d</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2199" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-317-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2595" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2596" class="mrow"><span id="MathJax-Span-2597" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.53em, 4.148em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2598" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2599" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-317" type="math/tex">d_i</script></span> is the order of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2200" class="MJXp-math"><span id="MJXp-Span-2201" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2202" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2203" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-318-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2600" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2601" class="mrow"><span id="MathJax-Span-2602" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2603" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2604" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-318" type="math/tex">g_i</script></span>,</p>
-
-<p>* <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2204" class="MJXp-math"><span id="MJXp-Span-2205" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2206" class="MJXp-mi MJXp-italic">α</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2207" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-319-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2605" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2606" class="mrow"><span id="MathJax-Span-2607" class="msubsup"><span style="display: inline-block; position: relative; width: 0.824em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.55em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2608" class="mi">α</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.552em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2609" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-319" type="math/tex">\alpha_i</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2208" class="MJXp-math"><span id="MJXp-Span-2209" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2210" class="MJXp-mi MJXp-italic">β</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2211" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-320-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2610" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2611" class="mrow"><span id="MathJax-Span-2612" class="msubsup"><span style="display: inline-block; position: relative; width: 0.778em; height: 0px;"><span style="position: absolute; clip: rect(3.168em, 1000.51em, 4.34em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2613" class="mi">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.506em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2614" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-320" type="math/tex">\beta_i</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2212" class="MJXp-math"><span id="MJXp-Span-2213" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2214" class="MJXp-mi MJXp-italic">γ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2215" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2216" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-2217" class="MJXp-mi MJXp-italic">s</span></span></span></span></span><span style="" tabindex="0" id="MathJax-Element-321-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2615" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2616" class="mrow"><span id="MathJax-Span-2617" class="msubsup"><span style="display: inline-block; position: relative; width: 1.051em; height: 0px;"><span style="position: absolute; clip: rect(3.411em, 1000.44em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2618" class="mi">γ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.028em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.41em;"><span id="MathJax-Span-2619" class="texatom"><span id="MathJax-Span-2620" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2621" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2622" class="mi">s</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-321" type="math/tex">\gamma_{rs}</script></span> are integers, and</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2218" class="MJXp-math MJXp-display"><span id="MJXp-Span-2219" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2220" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2221" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2222" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2223" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2224" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2225" class="MJXp-mo">=</span><span id="MJXp-Span-2226" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2227" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2228" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2229" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2230" class="MJXp-mi MJXp-italic">α</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2231" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2232" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-322-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-2623" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2624" class="mrow"><span id="MathJax-Span-2625" class="msubsup"><span style="display: inline-block; position: relative; width: 1.075em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2626" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span id="MathJax-Span-2627" class="texatom"><span id="MathJax-Span-2628" class="mrow"><span id="MathJax-Span-2629" class="msubsup"><span style="display: inline-block; position: relative; width: 0.546em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.33em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2630" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.82em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2631" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2632" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2633" class="msubsup"><span style="display: inline-block; position: relative; width: 1.111em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2634" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span id="MathJax-Span-2635" class="texatom"><span id="MathJax-Span-2636" class="mrow"><span id="MathJax-Span-2637" class="msubsup"><span style="display: inline-block; position: relative; width: 0.582em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.39em, 4.142em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2638" class="mi">α</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.39em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2639" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2640" class="mo">.</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-322" type="math/tex; mode=display">
- \xi^{g_i} = \xi^{\alpha_i}.
- </script></p>
-
-<p><a id="X8310E96086509397" name="X8310E96086509397"></a></p>
-
-<h4>9.13 <span class="Heading">Idempotents given by subgroups</span></h4>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2233" class="MJXp-math"><span id="MJXp-Span-2234" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-323-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2641" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2642" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2643" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-323" type="math/tex">G</script></span> be a finite group and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2235" class="MJXp-math"><span id="MJXp-Span-2236" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-324-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2644" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2645" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2646" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-324" type="math/tex">F</script></span> a field whose characteristic does not divide the order of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2237" class="MJXp-math"><span id="MJXp-Span-2238" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-325-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2647" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2648" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2649" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-325" type="math/tex">G</script></span>. If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2239" class="MJXp-math"><span id="MJXp-Span-2240" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-326-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2650" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2651" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2652" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-326" type="math/tex">H</script></span> is a subgroup of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2241" class="MJXp-math"><span id="MJXp-Span-2242" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-327-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2653" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2654" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2655" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-327" type="math/tex">G</script></span> then set</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2243" class="MJXp-math MJXp-display"><span id="MJXp-Span-2244" class="MJXp-mrow"><span id="MJXp-Span-2245" class="MJXp-munderover"><span><span class="MJXp-over"><span style="margin-bottom: -0.92em;" class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-2247" class="MJXp-mo">ˆ</span></span><span class=""><span id="MJXp-Span-2246" class="MJXp-mi MJXp-italic">H</span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2248" class="MJXp-mo">=</span><span id="MJXp-Span-2249" class="MJXp-mrow"><span style="margin-left: 0.167em; margin-right: 0.167em;" id="MJXp-Span-2250" class="MJXp-mo">|</span></span><span id="MJXp-Span-2251" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-2252" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2253" class="MJXp-mrow"><span style="margin-left: 0.167em; margin-right: 0.167em;" id="MJXp-Span-2254" class="MJXp-mo">|</span></span><span style="vertical-align: 0.5em;" id="MJXp-Span-2255" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2256" class="MJXp-mo">−</span><span id="MJXp-Span-2257" class="MJXp-mn">1</span></span></span><span id="MJXp-Span-2258" class="MJXp-munderover"><span class=""><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-2259" class="MJXp-mo"><span class="MJXp-largeop">∑</span></span></span><span class=" MJXp-script"><span style="margin-left: 0px;" id="MJXp-Span-2260" class="MJXp-mrow"><span id="MJXp-Span-2261" class="MJXp-mi MJXp-italic">x</span><span id="MJXp-Span-2262" class="MJXp-mo">∈</span><span id="MJXp-Span-2263" class="MJXp-mi MJXp-italic">H</span></span></span></span><span id="MJXp-Span-2264" class="MJXp-mi MJXp-italic">x</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2265" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-328-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-2656" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2657" class="mrow"><span id="MathJax-Span-2658" class="texatom"><span id="MathJax-Span-2659" class="mrow"><span id="MathJax-Span-2660" class="munderover"><span style="display: inline-block; position: relative; width: 1.035em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.77em, 4.135em, -1000em); top: -3.99em; left: 0.105em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2661" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.069em, 1000.98em, 3.571em, -1000em); top: -4.2em; left: 0.056em;"><span style="" id="MathJax-Span-2662" class="mo"><span style="font-family: STIXSizeTwoSym;">ˆ</span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2663" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2664" class="texatom"><span id="MathJax-Span-2665" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2666" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2667" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span id="MathJax-Span-2668" class="msubsup"><span style="display: inline-block; position: relative; width: 1.113em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.13em, 4.149em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-2669" class="texatom"><span id="MathJax-Span-2670" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2671" class="mo">|</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.2em;"><span id="MathJax-Span-2672" class="texatom"><span id="MathJax-Span-2673" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2674" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2675" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-2676" class="munderover"><span style="display: inline-block; position: relative; width: 1.344em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0.026em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-2677" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1001.34em, 4.254em, -1000em); top: -2.84em; left: 0em;"><span id="MathJax-Span-2678" class="texatom"><span id="MathJax-Span-2679" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2680" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2681" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2682" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.033em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2683" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2684" class="mo">.</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-328" type="math/tex; mode=display">
-\widehat{H} = |H|^{-1}\sum_{x \in H} x.
-</script></p>
-
-<p>The element <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2266" class="MJXp-math"><span id="MJXp-Span-2267" class="MJXp-mrow"><span id="MJXp-Span-2268" class="MJXp-munderover"><span><span class="MJXp-over"><span style="margin-bottom: -0.92em;" class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-2270" class="MJXp-mo">ˆ</span></span><span class=""><span id="MJXp-Span-2269" class="MJXp-mi MJXp-italic">H</span></span></span></span></span></span></span></span><span style="" tabindex="0" id="MathJax-Element-329-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2685" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2686" class="mrow"><span id="MathJax-Span-2687" class="texatom"><span id="MathJax-Span-2688" class="mrow"><span id="MathJax-Span-2689" class="munderover"><span style="display: inline-block; position: relative; width: 1.035em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.77em, 4.135em, -1000em); top: -3.99em; left: 0.105em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2690" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.069em, 1000.98em, 3.571em, -1000em); top: -4.2em; left: 0.056em;"><span style="" id="MathJax-Span-2691" class="mo"><span style="font-family: STIXSizeTwoSym;">ˆ</span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-329" type="math/tex">\widehat{H}</script></span> is an idempotent of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2271" class="MJXp-math"><span id="MJXp-Span-2272" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-2273" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-330-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2692" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2693" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2694" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2695" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-330" type="math/tex">FG</script></span> which is central in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2274" class="MJXp-math"><span id="MJXp-Span-2275" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-2276" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-331-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2696" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2697" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2698" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2699" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-331" type="math/tex">FG</script></span> if and only if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2277" class="MJXp-math"><span id="MJXp-Span-2278" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-332-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2700" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2701" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2702" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-332" type="math/tex">H</script></span> is normal in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2279" class="MJXp-math"><span id="MJXp-Span-2280" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-333-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2703" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2704" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2705" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-333" type="math/tex">G</script></span>.</p>
-
-<p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2281" class="MJXp-math"><span id="MJXp-Span-2282" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-334-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2706" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2707" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2708" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-334" type="math/tex">H</script></span> is a proper normal subgroup of a subgroup <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2283" class="MJXp-math"><span id="MJXp-Span-2284" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-335-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2709" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2710" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2711" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-335" type="math/tex">K</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2285" class="MJXp-math"><span id="MJXp-Span-2286" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-336-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2712" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2713" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2714" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-336" type="math/tex">G</script></span> then set</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2287" class="MJXp-math MJXp-display"><span id="MJXp-Span-2288" class="MJXp-mi MJXp-italic">ε</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2289" class="MJXp-mo">(</span><span id="MJXp-Span-2290" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2291" class="MJXp-mo">,</span><span id="MJXp-Span-2292" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2293" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2294" class="MJXp-mo">=</span><span id="MJXp-Span-2295" class="MJXp-munderover"><span class=""><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-2296" class="MJXp-mo"><span class="MJXp-largeop">âˆ</span></span></span><span class=" MJXp-script"><span style="margin-left: 0px;" id="MJXp-Span-2297" class="MJXp-mrow"><span id="MJXp-Span-2298" class="MJXp-mi MJXp-italic">L</span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2299" class="MJXp-mo">(</span><span id="MJXp-Span-2300" class="MJXp-mrow"><span id="MJXp-Span-2301" class="MJXp-munderover"><span><span class="MJXp-over"><span style="margin-bottom: -0.92em;" class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-2303" class="MJXp-mo">ˆ</span></span><span class=""><span id="MJXp-Span-2302" class="MJXp-mi MJXp-italic">N</span></span></span></span></span></span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-2304" class="MJXp-mo">−</span><span id="MJXp-Span-2305" class="MJXp-mrow"><span id="MJXp-Span-2306" class="MJXp-munderover"><span><span class="MJXp-over"><span style="margin-bottom: -0.92em;" class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-2308" class="MJXp-mo">ˆ</span></span><span class=""><span id="MJXp-Span-2307" class="MJXp-mi MJXp-italic">L</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2309" class="MJXp-mo">)</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-337-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-2715" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2716" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2717" class="mi">ε</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2718" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2719" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2720" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2721" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2722" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2723" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2724" class="munderover"><span style="display: inline-block; position: relative; width: 1.355em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.3em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-2725" class="mo">âˆ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.4em, 4.235em, -1000em); top: -2.84em; left: 0.48em;"><span id="MathJax-Span-2726" class="texatom"><span id="MathJax-Span-2727" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2728" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2729" class="mo">(</span><span id="MathJax-Span-2730" class="texatom"><span id="MathJax-Span-2731" class="mrow"><span id="MathJax-Span-2732" class="munderover"><span style="display: inline-block; position: relative; width: 1.043em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0.126em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2733" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.069em, 1000.98em, 3.571em, -1000em); top: -4.2em; left: 0.064em;"><span style="" id="MathJax-Span-2734" class="mo"><span style="font-family: STIXSizeTwoSym;">ˆ</span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2735" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-2736" class="texatom"><span id="MathJax-Span-2737" class="mrow"><span id="MathJax-Span-2738" class="munderover"><span style="display: inline-block; position: relative; width: 0.587em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2739" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.079em, 1000.56em, 3.581em, -1000em); top: -4.21em; left: 0.027em;"><span style="" id="MathJax-Span-2740" class="mo"><span style="font-family: STIXSizeOneSym;">ˆ</span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2741" class="mo">)</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-337" type="math/tex; mode=display"> \varepsilon(K,H) = \prod_{L} (\widehat{N}-\widehat{L})
-</script></p>
-
-<p>where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2310" class="MJXp-math"><span id="MJXp-Span-2311" class="MJXp-mi MJXp-italic">L</span></span></span><span style="" tabindex="0" id="MathJax-Element-338-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2742" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2743" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2744" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-338" type="math/tex">L</script></span> runs on the normal subgroups of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2312" class="MJXp-math"><span id="MJXp-Span-2313" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-339-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2745" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2746" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2747" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-339" type="math/tex">K</script></span> which are minimal among the normal subgroups of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2314" class="MJXp-math"><span id="MJXp-Span-2315" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-340-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2748" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2749" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2750" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-340" type="math/tex">K</script></span> containing <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2316" class="MJXp-math"><span id="MJXp-Span-2317" class="MJXp-mi MJXp-italic">N</span></span></span><span style="" tabindex="0" id="MathJax-Element-341-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2751" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2752" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2753" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-341" type="math/tex">N</script></span> properly. By convention, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2318" class="MJXp-math"><span id="MJXp-Span-2319" class="MJXp-mi MJXp-italic">ε</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2320" class="MJXp-mo">(</span><span id="MJXp-Span-2321" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2322" class="MJXp-mo">,</span><span id="MJXp-Span-2323" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2324" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2325" class="MJXp-mo">=</span><span id="MJXp-Span-2326" class="MJXp-mrow"><span id="MJXp-Span-2327" class="MJXp-munderover"><span><span class="MJXp-over"><span style="margin-bottom: -0.92em;" class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-2329" class="MJXp-mo">ˆ</span></span><span class=""><span id="MJXp-Span-2328" class="MJXp-mi MJXp-italic">K</span></span></span></span></span></span></span></span><span style="" tabindex="0" id="MathJax-Element-342-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2754" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2755" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2756" class="mi">ε</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2757" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2758" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2759" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2760" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2761" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2762" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-2763" class="texatom"><span id="MathJax-Span-2764" class="mrow"><span id="MathJax-Span-2765" class="munderover"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0.129em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2766" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.069em, 1000.98em, 3.571em, -1000em); top: -4.2em; left: 0.061em;"><span style="" id="MathJax-Span-2767" class="mo"><span style="font-family: STIXSizeTwoSym;">ˆ</span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-342" type="math/tex">\varepsilon(K,K)=\widehat{K}</script></span>. The element <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2330" class="MJXp-math"><span id="MJXp-Span-2331" class="MJXp-mi MJXp-italic">ε</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2332" class="MJXp-mo">(</span><span id="MJXp-Span-2333" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2334" class="MJXp-mo">,</span><span id="MJXp-Span-2335" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2336" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-343-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2768" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2769" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2770" class="mi">ε</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2771" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2772" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2773" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2774" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2775" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-343" type="math/tex">\varepsilon(K,H)</script></span> is an idempotent of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2337" class="MJXp-math"><span id="MJXp-Span-2338" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-2339" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-344-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2776" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2777" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2778" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2779" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-344" type="math/tex">FG</script></span>.</p>
-
-<p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2340" class="MJXp-math"><span id="MJXp-Span-2341" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-345-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2780" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2781" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2782" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-345" type="math/tex">H</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2342" class="MJXp-math"><span id="MJXp-Span-2343" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-346-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2783" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2784" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2785" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-346" type="math/tex">K</script></span> are subgroups of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2344" class="MJXp-math"><span id="MJXp-Span-2345" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-347-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2786" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2787" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2788" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-347" type="math/tex">G</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2346" class="MJXp-math"><span id="MJXp-Span-2347" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-348-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2789" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2790" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2791" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-348" type="math/tex">H</script></span> is normal in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2348" class="MJXp-math"><span id="MJXp-Span-2349" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-349-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2792" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2793" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2794" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-349" type="math/tex">K</script></span> then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2350" class="MJXp-math"><span id="MJXp-Span-2351" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2352" class="MJXp-mo">(</span><span id="MJXp-Span-2353" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2354" class="MJXp-mo">,</span><span id="MJXp-Span-2355" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2356" class="MJXp-mo">,</span><span id="MJXp-Span-2357" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2358" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-350-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2795" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2796" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2797" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2798" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2799" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2800" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2801" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2802" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2803" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2804" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-350" type="math/tex">e(G,K,H)</script></span> denotes the sum of all different <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2359" class="MJXp-math"><span id="MJXp-Span-2360" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-351-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2805" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2806" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2807" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-351" type="math/tex">G</script></span>-conjugates of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2361" class="MJXp-math"><span id="MJXp-Span-2362" class="MJXp-mi MJXp-italic">ε</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2363" class="MJXp-mo">(</span><span id="MJXp-Span-2364" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2365" class="MJXp-mo">,</span><span id="MJXp-Span-2366" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2367" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-352-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2808" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2809" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2810" class="mi">ε</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2811" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2812" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2813" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2814" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2815" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-352" type="math/tex">\varepsilon(K,H)</script></span>. The element <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2368" class="MJXp-math"><span id="MJXp-Span-2369" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2370" class="MJXp-mo">(</span><span id="MJXp-Span-2371" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2372" class="MJXp-mo">,</span><span id="MJXp-Span-2373" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2374" class="MJXp-mo">,</span><span id="MJXp-Span-2375" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2376" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-353-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2816" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2817" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2818" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2819" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2820" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2821" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2822" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2823" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2824" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2825" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-353" type="math/tex">e(G,K,H)</script></span> is central in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2377" class="MJXp-math"><span id="MJXp-Span-2378" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-2379" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-354-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2826" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2827" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2828" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2829" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-354" type="math/tex">FG</script></span>. In general it is not an idempotent but if the different conjugates of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2380" class="MJXp-math"><span id="MJXp-Span-2381" class="MJXp-mi MJXp-italic">ε</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2382" class="MJXp-mo">(</span><span id="MJXp-Span-2383" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2384" class="MJXp-mo">,</span><span id="MJXp-Span-2385" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2386" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-355-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2830" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2831" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2832" class="mi">ε</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2833" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2834" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2835" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2836" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2837" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-355" type="math/tex">\varepsilon(K,H)</script></span> are orthogonal then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2387" class="MJXp-math"><span id="MJXp-Span-2388" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2389" class="MJXp-mo">(</span><span id="MJXp-Span-2390" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2391" class="MJXp-mo">,</span><span id="MJXp-Span-2392" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2393" class="MJXp-mo">,</span><span id="MJXp-Span-2394" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2395" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-356-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2838" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2839" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2840" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2841" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2842" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2843" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2844" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2845" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2846" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2847" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-356" type="math/tex">e(G,K,H)</script></span> is a central idempotent of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2396" class="MJXp-math"><span id="MJXp-Span-2397" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-2398" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-357-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2848" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2849" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2850" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2851" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-357" type="math/tex">FG</script></span>.</p>
-
-<p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2399" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2400" class="MJXp-mo">(</span><span id="MJXp-Span-2401" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2402" class="MJXp-mo">,</span><span id="MJXp-Span-2403" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2404" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-358-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2852" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2853" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2854" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2855" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2856" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2857" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2858" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-358" type="math/tex">(K,H)</script></span> is a Shoda Pair (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7D518BAB80EDE190"><span class="RefLink">9.14</span></a>) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2405" class="MJXp-math"><span id="MJXp-Span-2406" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-359-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2859" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2860" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2861" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-359" type="math/tex">G</script></span> then there is a non-zero rational number <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2407" class="MJXp-math"><span id="MJXp-Span-2408" class="MJXp-mi MJXp-italic">a</span></span></span><span style="" tabindex="0" id="MathJax-Element-360-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2862" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2863" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2864" class="mi">a</span></span></span></span></span></nobr></span><script id="MathJax-Element-360" type="math/tex">a</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2409" class="MJXp-math"><span id="MJXp-Span-2410" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-2411" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2412" class="MJXp-mo">(</span><span id="MJXp-Span-2413" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2414" class="MJXp-mo">,</span><span id="MJXp-Span-2415" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2416" class="MJXp-mo">,</span><span id="MJXp-Span-2417" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2418" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2419" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-361-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2865" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2866" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2867" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2868" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2869" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2870" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2871" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2872" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2873" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2874" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2875" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2876" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-361" type="math/tex">ae(G,K,H))</script></span> is a <em>primitive central idempotent</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of the rational group algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2420" class="MJXp-math"><span id="MJXp-Span-2421" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2422" class="MJXp-mo">â„š</span></span><span id="MJXp-Span-2423" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-362-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2877" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2878" class="mrow"><span id="MathJax-Span-2879" class="texatom"><span id="MathJax-Span-2880" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2881" class="mo">â„š</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2882" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-362" type="math/tex">â„š G</script></span>. If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2424" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2425" class="MJXp-mo">(</span><span id="MJXp-Span-2426" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2427" class="MJXp-mo">,</span><span id="MJXp-Span-2428" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2429" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-363-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2883" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2884" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2885" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2886" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2887" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2888" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2889" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-363" type="math/tex">(K,H)</script></span> is a strong Shoda pair (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2430" class="MJXp-math"><span id="MJXp-Span-2431" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-364-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2890" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2891" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2892" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-364" type="math/tex">G</script></span> then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2432" class="MJXp-math"><span id="MJXp-Span-2433" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2434" class="MJXp-mo">(</span><span id="MJXp-Span-2435" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2436" class="MJXp-mo">,</span><span id="MJXp-Span-2437" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2438" class="MJXp-mo">,</span><span id="MJXp-Span-2439" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2440" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-365-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2893" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2894" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2895" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2896" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2897" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2898" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2899" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2900" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2901" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2902" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-365" type="math/tex">e(G,K,H)</script></span> is a primitive central idempotent of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2441" class="MJXp-math"><span id="MJXp-Span-2442" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2443" class="MJXp-mo">â„š</span></span><span id="MJXp-Span-2444" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-366-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2903" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2904" class="mrow"><span id="MathJax-Span-2905" class="texatom"><span id="MathJax-Span-2906" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2907" class="mo">â„š</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2908" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-366" type="math/tex">â„š G</script></span>.</p>
-
-<p>Assume now that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2445" class="MJXp-math"><span id="MJXp-Span-2446" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-367-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2909" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2910" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2911" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-367" type="math/tex">F</script></span> is a finite field of order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2447" class="MJXp-math"><span id="MJXp-Span-2448" class="MJXp-mi MJXp-italic">q</span></span></span><span style="" tabindex="0" id="MathJax-Element-368-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2912" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2913" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2914" class="mi">q</span></span></span></span></span></nobr></span><script id="MathJax-Element-368" type="math/tex">q</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2449" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2450" class="MJXp-mo">(</span><span id="MJXp-Span-2451" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2452" class="MJXp-mo">,</span><span id="MJXp-Span-2453" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2454" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-369-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2915" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2916" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2917" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2918" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2919" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2920" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2921" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-369" type="math/tex">(K,H)</script></span> is a strong Shoda pair of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2455" class="MJXp-math"><span id="MJXp-Span-2456" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-370-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2922" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2923" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2924" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-370" type="math/tex">G</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2457" class="MJXp-math"><span id="MJXp-Span-2458" class="MJXp-mi MJXp-italic">C</span></span></span><span style="" tabindex="0" id="MathJax-Element-371-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2925" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2926" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2927" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-371" type="math/tex">C</script></span> is a cyclotomic class of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2459" class="MJXp-math"><span id="MJXp-Span-2460" class="MJXp-mi MJXp-italic">K</span><span id="MJXp-Span-2461" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2462" class="MJXp-mo">/</span></span><span id="MJXp-Span-2463" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-372-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2928" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2929" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2930" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span id="MathJax-Span-2931" class="texatom"><span id="MathJax-Span-2932" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2933" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2934" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-372" type="math/tex">K/H</script></span> containing a generator of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2464" class="MJXp-math"><span id="MJXp-Span-2465" class="MJXp-mi MJXp-italic">K</span><span id="MJXp-Span-2466" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2467" class="MJXp-mo">/</span></span><span id="MJXp-Span-2468" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-373-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2935" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2936" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2937" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span id="MathJax-Span-2938" class="texatom"><span id="MathJax-Span-2939" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2940" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2941" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-373" type="math/tex">K/H</script></span>. Then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2469" class="MJXp-math"><span id="MJXp-Span-2470" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2471" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2472" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2473" class="MJXp-mo">(</span><span id="MJXp-Span-2474" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2475" class="MJXp-mo">,</span><span id="MJXp-Span-2476" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2477" class="MJXp-mo">,</span><span id="MJXp-Span-2478" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2479" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-374-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2942" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2943" class="mrow"><span id="MathJax-Span-2944" class="msubsup"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2945" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2946" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2947" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2948" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2949" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2950" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2951" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2952" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2953" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-374" type="math/tex">e_C(G,K,H)</script></span> is a primitive central idempotent of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2480" class="MJXp-math"><span id="MJXp-Span-2481" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-2482" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-375-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2954" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2955" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2956" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2957" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-375" type="math/tex">FG</script></span> (see <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.17</span></a>).</p>
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-<p><a id="X7D518BAB80EDE190" name="X7D518BAB80EDE190"></a></p>
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-<h4>9.14 <span class="Heading">Shoda pairs of a group</span></h4>
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-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2483" class="MJXp-math"><span id="MJXp-Span-2484" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-376-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2958" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2959" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2960" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-376" type="math/tex">G</script></span> be a finite group. A <em>Shoda pair</em> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2485" class="MJXp-math"><span id="MJXp-Span-2486" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-377-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2961" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2962" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2963" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-377" type="math/tex">G</script></span> is a pair <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2487" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2488" class="MJXp-mo">(</span><span id="MJXp-Span-2489" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2490" class="MJXp-mo">,</span><span id="MJXp-Span-2491" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2492" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-378-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2964" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2965" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2966" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2967" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2968" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2969" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2970" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-378" type="math/tex">(K,H)</script></span> of subgroups of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2493" class="MJXp-math"><span id="MJXp-Span-2494" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-379-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2971" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2972" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2973" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-379" type="math/tex">G</script></span> for which there is a linear character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2495" class="MJXp-math"><span id="MJXp-Span-2496" class="MJXp-mi MJXp-italic">χ</span></span></span><span style="" tabindex="0" id="MathJax-Element-380-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2974" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2975" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2976" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-380" type="math/tex">\chi</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2497" class="MJXp-math"><span id="MJXp-Span-2498" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-381-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2977" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2978" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2979" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-381" type="math/tex">K</script></span> with kernel <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2499" class="MJXp-math"><span id="MJXp-Span-2500" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-382-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2980" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2981" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2982" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-382" type="math/tex">H</script></span> such that the induced character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2501" class="MJXp-math"><span id="MJXp-Span-2502" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2503" class="MJXp-mi MJXp-italic">χ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2504" class="MJXp-mi MJXp-italic MJXp-script">G</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-383-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2983" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2984" class="mrow"><span id="MathJax-Span-2985" class="msubsup"><span style="display: inline-block; position: relative; width: 1.146em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.5em, 4.342em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2986" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.56em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2987" class="mi">G</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-383" type="math/tex">\chi^G</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2505" class="MJXp-math"><span id="MJXp-Span-2506" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-384-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2988" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2989" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2990" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-384" type="math/tex">G</script></span> is irreducible. By <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBS">[Sho33]</a> or <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBORS">[ORS04]</a>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2507" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2508" class="MJXp-mo">(</span><span id="MJXp-Span-2509" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2510" class="MJXp-mo">,</span><span id="MJXp-Span-2511" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2512" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-385-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2991" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2992" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2993" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2994" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2995" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2996" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2997" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-385" type="math/tex">(K,H)</script></span> is a Shoda pair if and only if the following conditions hold:</p>
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-<p>* <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2513" class="MJXp-math"><span id="MJXp-Span-2514" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-386-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-2998" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-2999" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3000" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-386" type="math/tex">H</script></span> is normal in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2515" class="MJXp-math"><span id="MJXp-Span-2516" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-387-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3001" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3002" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3003" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-387" type="math/tex">K</script></span>,</p>
-
-<p>* <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2517" class="MJXp-math"><span id="MJXp-Span-2518" class="MJXp-mi MJXp-italic">K</span><span id="MJXp-Span-2519" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2520" class="MJXp-mo">/</span></span><span id="MJXp-Span-2521" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-388-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3004" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3005" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3006" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span id="MathJax-Span-3007" class="texatom"><span id="MathJax-Span-3008" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3009" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3010" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-388" type="math/tex">K/H</script></span> is cyclic and</p>
-
-<p>* if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2522" class="MJXp-math"><span id="MJXp-Span-2523" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2524" class="MJXp-mi MJXp-italic">K</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2525" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-2526" class="MJXp-mo">∩</span><span id="MJXp-Span-2527" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2528" class="MJXp-mo">⊆</span><span id="MJXp-Span-2529" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-389-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3011" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3012" class="mrow"><span id="MathJax-Span-3013" class="msubsup"><span style="display: inline-block; position: relative; width: 1.217em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3014" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.788em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3015" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3016" class="mo">∩</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3017" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3018" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3019" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-389" type="math/tex">K^g \cap K \subseteq H</script></span> for some <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2530" class="MJXp-math"><span id="MJXp-Span-2531" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2532" class="MJXp-mo">∈</span><span id="MJXp-Span-2533" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-390-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3020" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3021" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3022" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3023" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3024" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-390" type="math/tex">g \in G</script></span> then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2534" class="MJXp-math"><span id="MJXp-Span-2535" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2536" class="MJXp-mo">∈</span><span id="MJXp-Span-2537" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-391-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3025" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3026" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3027" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3028" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3029" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-391" type="math/tex">g \in K</script></span>.</p>
-
-<p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2538" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2539" class="MJXp-mo">(</span><span id="MJXp-Span-2540" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2541" class="MJXp-mo">,</span><span id="MJXp-Span-2542" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2543" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-392-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3030" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3031" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3032" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3033" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3034" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3035" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3036" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-392" type="math/tex">(K,H)</script></span> is a Shoda pair and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2544" class="MJXp-math"><span id="MJXp-Span-2545" class="MJXp-mi MJXp-italic">χ</span></span></span><span style="" tabindex="0" id="MathJax-Element-393-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3037" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3038" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3039" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-393" type="math/tex">\chi</script></span> is a linear character of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2546" class="MJXp-math"><span id="MJXp-Span-2547" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2548" class="MJXp-mo">≤</span><span id="MJXp-Span-2549" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-394-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3040" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3041" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3042" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3043" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3044" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-394" type="math/tex">K\le G</script></span> with kernel <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2550" class="MJXp-math"><span id="MJXp-Span-2551" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-395-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3045" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3046" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3047" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-395" type="math/tex">H</script></span> then the <em>primitive central idempotent</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2552" class="MJXp-math"><span id="MJXp-Span-2553" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2554" class="MJXp-mo">ℚ</span></span><span id="MJXp-Span-2555" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-396-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3048" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3049" class="mrow"><span id="MathJax-Span-3050" class="texatom"><span id="MathJax-Span-3051" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3052" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3053" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-396" type="math/tex">ℚ G</script></span> associated to the irreducible character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2556" class="MJXp-math"><span id="MJXp-Span-2557" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2558" class="MJXp-mi MJXp-italic">χ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2559" class="MJXp-mi MJXp-italic MJXp-script">G</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-397-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3054" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3055" class="mrow"><span id="MathJax-Span-3056" class="msubsup"><span style="display: inline-block; position: relative; width: 1.146em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.5em, 4.342em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3057" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.56em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3058" class="mi">G</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-397" type="math/tex">\chi^G</script></span> is of the form <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2560" class="MJXp-math"><span id="MJXp-Span-2561" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2562" class="MJXp-mo">=</span><span id="MJXp-Span-2563" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2564" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2565" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2566" class="MJXp-mo">ℚ</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2567" class="MJXp-mo">(</span><span id="MJXp-Span-2568" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2569" class="MJXp-mi MJXp-italic">χ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2570" class="MJXp-mi MJXp-italic MJXp-script">G</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2571" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2572" class="MJXp-mo">=</span><span id="MJXp-Span-2573" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-2574" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2575" class="MJXp-mo">(</span><span id="MJXp-Span-2576" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2577" class="MJXp-mo">,</span><span id="MJXp-Span-2578" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2579" class="MJXp-mo">,</span><span id="MJXp-Span-2580" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2581" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-398-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3059" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3060" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3061" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3062" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-3063" class="msubsup"><span style="display: inline-block; position: relative; width: 1.06em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3064" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-3065" class="texatom"><span id="MathJax-Span-3066" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3067" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3068" class="mo">(</span><span id="MathJax-Span-3069" class="msubsup"><span style="display: inline-block; position: relative; width: 1.146em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.5em, 4.342em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3070" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.56em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3071" class="mi">G</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3072" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3073" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3074" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3075" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3076" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3077" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3078" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3079" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3080" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3081" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3082" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-398" type="math/tex">e=e_ℚ (\chi^G)=a e(G,K,H)</script></span> for some <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2582" class="MJXp-math"><span id="MJXp-Span-2583" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2584" class="MJXp-mo">∈</span><span id="MJXp-Span-2585" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2586" class="MJXp-mo">ℚ</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-399-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3083" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3084" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3085" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3086" class="mo">∈</span><span style="padding-left: 0.313em;" id="MathJax-Span-3087" class="texatom"><span id="MathJax-Span-3088" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3089" class="mo">ℚ</span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-399" type="math/tex">a \in ℚ </script></span> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBORS">[ORS04]</a> (see <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8310E96086509397"><span class="RefLink">9.13</span></a> for the definition of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2587" class="MJXp-math"><span id="MJXp-Span-2588" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2589" class="MJXp-mo">(</span><span id="MJXp-Span-2590" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2591" class="MJXp-mo">,</span><span id="MJXp-Span-2592" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2593" class="MJXp-mo">,</span><span id="MJXp-Span-2594" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2595" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-400-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3090" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3091" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3092" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3093" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3094" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3095" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3096" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3097" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3098" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3099" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-400" type="math/tex">e(G,K,H)</script></span>). In that case we say that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2596" class="MJXp-math"><span id="MJXp-Span-2597" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-401-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3100" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3101" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3102" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-401" type="math/tex">e</script></span> is the <em>primitive central idempotent realized by the Shoda pair</em> <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2598" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2599" class="MJXp-mo">(</span><span id="MJXp-Span-2600" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2601" class="MJXp-mo">,</span><span id="MJXp-Span-2602" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2603" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-402-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3103" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3104" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3105" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3106" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3107" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3108" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3109" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-402" type="math/tex">(K,H)</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2604" class="MJXp-math"><span id="MJXp-Span-2605" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-403-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3110" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3111" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3112" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-403" type="math/tex">G</script></span>.</p>
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-<p>A group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2606" class="MJXp-math"><span id="MJXp-Span-2607" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-404-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3113" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3114" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3115" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-404" type="math/tex">G</script></span> is monomial, that is every irreducible character of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2608" class="MJXp-math"><span id="MJXp-Span-2609" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-405-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3116" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3117" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3118" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-405" type="math/tex">G</script></span> is monomial, if and only if every primitive central idempotent of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2610" class="MJXp-math"><span id="MJXp-Span-2611" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2612" class="MJXp-mo">â„š</span></span><span id="MJXp-Span-2613" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-406-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3119" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3120" class="mrow"><span id="MathJax-Span-3121" class="texatom"><span id="MathJax-Span-3122" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3123" class="mo">â„š</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3124" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-406" type="math/tex">â„š G</script></span> is realizable by a Shoda pair of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2614" class="MJXp-math"><span id="MJXp-Span-2615" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-407-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3125" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3126" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3127" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-407" type="math/tex">G</script></span>.</p>
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-<p><a id="X7E3479527BAE5B9E" name="X7E3479527BAE5B9E"></a></p>
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-<h4>9.15 <span class="Heading">Strong Shoda pairs of a group</span></h4>
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-<p>A <em>strong Shoda pair</em> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2616" class="MJXp-math"><span id="MJXp-Span-2617" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-408-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3128" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3129" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3130" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-408" type="math/tex">G</script></span> is a pair <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2618" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2619" class="MJXp-mo">(</span><span id="MJXp-Span-2620" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2621" class="MJXp-mo">,</span><span id="MJXp-Span-2622" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2623" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-409-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3131" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3132" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3133" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3134" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3135" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3136" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3137" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-409" type="math/tex">(K,H)</script></span> of subgroups of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2624" class="MJXp-math"><span id="MJXp-Span-2625" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-410-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3138" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3139" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3140" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-410" type="math/tex">G</script></span> satisfying the following conditions:</p>
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-<p>* <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2626" class="MJXp-math"><span id="MJXp-Span-2627" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-411-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3141" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3142" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3143" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-411" type="math/tex">H</script></span> is normal in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2628" class="MJXp-math"><span id="MJXp-Span-2629" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-412-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3144" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3145" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3146" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-412" type="math/tex">K</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2630" class="MJXp-math"><span id="MJXp-Span-2631" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-413-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3147" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3148" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3149" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-413" type="math/tex">K</script></span> is normal in the normalizer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2632" class="MJXp-math"><span id="MJXp-Span-2633" class="MJXp-mi MJXp-italic">N</span></span></span><span style="" tabindex="0" id="MathJax-Element-414-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3150" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3151" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3152" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-414" type="math/tex">N</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2634" class="MJXp-math"><span id="MJXp-Span-2635" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-415-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3153" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3154" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3155" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-415" type="math/tex">H</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2636" class="MJXp-math"><span id="MJXp-Span-2637" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-416-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3156" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3157" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3158" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-416" type="math/tex">G</script></span>,</p>
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-<p>* <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2638" class="MJXp-math"><span id="MJXp-Span-2639" class="MJXp-mi MJXp-italic">K</span><span id="MJXp-Span-2640" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2641" class="MJXp-mo">/</span></span><span id="MJXp-Span-2642" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-417-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3159" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3160" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3161" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span id="MathJax-Span-3162" class="texatom"><span id="MathJax-Span-3163" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3164" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3165" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-417" type="math/tex">K/H</script></span> is cyclic and a maximal abelian subgroup of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2643" class="MJXp-math"><span id="MJXp-Span-2644" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-2645" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2646" class="MJXp-mo">/</span></span><span id="MJXp-Span-2647" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-418-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3166" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3167" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3168" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3169" class="texatom"><span id="MathJax-Span-3170" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3171" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3172" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-418" type="math/tex">N/H</script></span> and</p>
-
-<p>* for every <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2648" class="MJXp-math"><span id="MJXp-Span-2649" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2650" class="MJXp-mo">∈</span><span id="MJXp-Span-2651" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-2652" class="MJXp-mo">∖</span><span id="MJXp-Span-2653" class="MJXp-mi MJXp-italic">N</span></span></span><span style="" tabindex="0" id="MathJax-Element-419-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3173" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3174" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3175" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3176" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3177" class="mi">G</span><span style="font-family: STIXVariants; padding-left: 0.25em;" id="MathJax-Span-3178" class="mo">∖</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3179" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-419" type="math/tex">g \in G\setminus N</script></span> , <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2654" class="MJXp-math"><span id="MJXp-Span-2655" class="MJXp-mi MJXp-italic">ε</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2656" class="MJXp-mo">(</span><span id="MJXp-Span-2657" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2658" class="MJXp-mo">,</span><span id="MJXp-Span-2659" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2660" class="MJXp-mo">)</span><span id="MJXp-Span-2661" class="MJXp-mi MJXp-italic">ε</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2662" class="MJXp-mo">(</span><span id="MJXp-Span-2663" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2664" class="MJXp-mo">,</span><span id="MJXp-Span-2665" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-2666" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-2667" class="MJXp-mo">)</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2668" class="MJXp-mi MJXp-italic MJXp-script">g</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2669" class="MJXp-mo">=</span><span id="MJXp-Span-2670" class="MJXp-mn">0</span></span></span><span style="" tabindex="0" id="MathJax-Element-420-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3180" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3181" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3182" class="mi">ε</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3183" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3184" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3185" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3186" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3187" class="mo">)</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3188" class="mi">ε</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3189" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3190" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3191" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3192" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span id="MathJax-Span-3193" class="msubsup"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-3194" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3195" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3196" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3197" class="mn">0</span></span></span></span></span></nobr></span><script id="MathJax-Element-420" type="math/tex">\varepsilon(K,H)\varepsilon(K,H)^g=0</script></span>. (See <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8310E96086509397"><span class="RefLink">9.13</span></a> for the definition of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2671" class="MJXp-math"><span id="MJXp-Span-2672" class="MJXp-mi MJXp-italic">ε</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2673" class="MJXp-mo">(</span><span id="MJXp-Span-2674" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2675" class="MJXp-mo">,</span><span id="MJXp-Span-2676" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2677" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-421-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3198" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3199" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3200" class="mi">ε</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3201" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3202" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3203" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3204" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3205" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-421" type="math/tex">\varepsilon(K,H)</script></span>).</p>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2678" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2679" class="MJXp-mo">(</span><span id="MJXp-Span-2680" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2681" class="MJXp-mo">,</span><span id="MJXp-Span-2682" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2683" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-422-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3206" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3207" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3208" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3209" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3210" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3211" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3212" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-422" type="math/tex">(K,H)</script></span> be a strong Shoda pair of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2684" class="MJXp-math"><span id="MJXp-Span-2685" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-423-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3213" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3214" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3215" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-423" type="math/tex">G</script></span>. Then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2686" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2687" class="MJXp-mo">(</span><span id="MJXp-Span-2688" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2689" class="MJXp-mo">,</span><span id="MJXp-Span-2690" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2691" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-424-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3216" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3217" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3218" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3219" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3220" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3221" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3222" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-424" type="math/tex">(K,H)</script></span> is a Shoda pair (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7D518BAB80EDE190"><span class="RefLink">9.14</span></a>) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2692" class="MJXp-math"><span id="MJXp-Span-2693" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-425-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3223" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3224" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3225" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-425" type="math/tex">G</script></span>. Thus there is a linear character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2694" class="MJXp-math"><span id="MJXp-Span-2695" class="MJXp-mi MJXp-italic">θ</span></span></span><span style="" tabindex="0" id="MathJax-Element-426-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3226" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3227" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3228" class="mi">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-426" type="math/tex">\theta</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2696" class="MJXp-math"><span id="MJXp-Span-2697" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-427-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3229" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3230" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3231" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-427" type="math/tex">K</script></span> with kernel <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2698" class="MJXp-math"><span id="MJXp-Span-2699" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-428-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3232" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3233" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3234" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-428" type="math/tex">H</script></span> such that the induced character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2700" class="MJXp-math"><span id="MJXp-Span-2701" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2702" class="MJXp-mo">=</span><span id="MJXp-Span-2703" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2704" class="MJXp-mo">(</span><span id="MJXp-Span-2705" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2706" class="MJXp-mo">,</span><span id="MJXp-Span-2707" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2708" class="MJXp-mo">,</span><span id="MJXp-Span-2709" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2710" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2711" class="MJXp-mo">=</span><span id="MJXp-Span-2712" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2713" class="MJXp-mi MJXp-italic">θ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2714" class="MJXp-mi MJXp-italic MJXp-script">G</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-429-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3235" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3236" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3237" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3238" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3239" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3240" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3241" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3242" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3243" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3244" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3245" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3246" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3247" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-3248" class="msubsup"><span style="display: inline-block; position: relative; width: 1.134em; height: 0px;"><span style="position: absolute; clip: rect(3.168em, 1000.49em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3249" class="mi">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.548em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3250" class="mi">G</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-429" type="math/tex">\chi=\chi(G,K,H)=\theta^G</script></span> is irreducible. Moreover the <em>primitive central idempotent</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2715" class="MJXp-math"><span id="MJXp-Span-2716" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2717" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2718" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2719" class="MJXp-mrow"><span id="MJXp-Span-2720" class="MJXp-mo">ℚ</span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2721" class="MJXp-mo">(</span><span id="MJXp-Span-2722" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2723" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-430-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3251" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3252" class="mrow"><span id="MathJax-Span-3253" class="msubsup"><span style="display: inline-block; position: relative; width: 1.06em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3254" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-3255" class="texatom"><span id="MathJax-Span-3256" class="mrow"><span id="MathJax-Span-3257" class="texatom"><span id="MathJax-Span-3258" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3259" class="mo">ℚ</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3260" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3261" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3262" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-430" type="math/tex">e_{ℚ }(\chi)</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2724" class="MJXp-math"><span id="MJXp-Span-2725" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2726" class="MJXp-mo">ℚ</span></span><span id="MJXp-Span-2727" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-431-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3263" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3264" class="mrow"><span id="MathJax-Span-3265" class="texatom"><span id="MathJax-Span-3266" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3267" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3268" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-431" type="math/tex">ℚ G</script></span> realized by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2728" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2729" class="MJXp-mo">(</span><span id="MJXp-Span-2730" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2731" class="MJXp-mo">,</span><span id="MJXp-Span-2732" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2733" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-432-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3269" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3270" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3271" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3272" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3273" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3274" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3275" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-432" type="math/tex">(K,H)</script></span> is <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2734" class="MJXp-math"><span id="MJXp-Span-2735" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2736" class="MJXp-mo">(</span><span id="MJXp-Span-2737" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2738" class="MJXp-mo">,</span><span id="MJXp-Span-2739" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2740" class="MJXp-mo">,</span><span id="MJXp-Span-2741" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2742" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-433-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3276" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3277" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3278" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3279" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3280" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3281" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3282" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3283" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3284" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3285" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-433" type="math/tex">e(G,K,H)</script></span>, see <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBORS">[ORS04]</a>.</p>
-
-<p>Two <em>strong Shoda pairs</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2743" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2744" class="MJXp-mo">(</span><span id="MJXp-Span-2745" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2746" class="MJXp-mi MJXp-italic">K</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2747" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2748" class="MJXp-mo">,</span><span id="MJXp-Span-2749" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2750" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2751" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2752" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-434-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3286" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3287" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3288" class="mo">(</span><span id="MathJax-Span-3289" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3290" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3291" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3292" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3293" class="msubsup"><span style="display: inline-block; position: relative; width: 1.151em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.77em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3294" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3295" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3296" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-434" type="math/tex">(K_1,H_1)</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2753" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2754" class="MJXp-mo">(</span><span id="MJXp-Span-2755" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2756" class="MJXp-mi MJXp-italic">K</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2757" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2758" class="MJXp-mo">,</span><span id="MJXp-Span-2759" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2760" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2761" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2762" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-435-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3297" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3298" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3299" class="mo">(</span><span id="MathJax-Span-3300" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3301" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3302" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3303" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3304" class="msubsup"><span style="display: inline-block; position: relative; width: 1.151em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.77em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3305" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3306" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3307" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-435" type="math/tex">(K_2,H_2)</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2763" class="MJXp-math"><span id="MJXp-Span-2764" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-436-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3308" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3309" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3310" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-436" type="math/tex">G</script></span> are said to be <em>equivalent</em> if the characters <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2765" class="MJXp-math"><span id="MJXp-Span-2766" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2767" class="MJXp-mo">(</span><span id="MJXp-Span-2768" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2769" class="MJXp-mo">,</span><span id="MJXp-Span-2770" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2771" class="MJXp-mi MJXp-italic">K</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2772" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2773" class="MJXp-mo">,</span><span id="MJXp-Span-2774" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2775" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2776" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2777" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-437-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3311" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3312" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3313" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3314" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3315" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3316" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3317" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3318" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3319" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3320" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3321" class="msubsup"><span style="display: inline-block; position: relative; width: 1.151em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.77em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3322" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3323" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3324" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-437" type="math/tex">\chi(G,K_1,H_1)</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2778" class="MJXp-math"><span id="MJXp-Span-2779" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2780" class="MJXp-mo">(</span><span id="MJXp-Span-2781" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2782" class="MJXp-mo">,</span><span id="MJXp-Span-2783" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2784" class="MJXp-mi MJXp-italic">K</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2785" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2786" class="MJXp-mo">,</span><span id="MJXp-Span-2787" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2788" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2789" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2790" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-438-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3325" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3326" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3327" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3328" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3329" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3330" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3331" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3332" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3333" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3334" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3335" class="msubsup"><span style="display: inline-block; position: relative; width: 1.151em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.77em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3336" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3337" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3338" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-438" type="math/tex">\chi(G,K_2,H_2)</script></span> are Galois conjugate, or equivalently if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2791" class="MJXp-math"><span id="MJXp-Span-2792" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2793" class="MJXp-mo">(</span><span id="MJXp-Span-2794" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2795" class="MJXp-mo">,</span><span id="MJXp-Span-2796" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2797" class="MJXp-mi MJXp-italic">K</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2798" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2799" class="MJXp-mo">,</span><span id="MJXp-Span-2800" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2801" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2802" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2803" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2804" class="MJXp-mo">=</span><span id="MJXp-Span-2805" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2806" class="MJXp-mo">(</span><span id="MJXp-Span-2807" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2808" class="MJXp-mo">,</span><span id="MJXp-Span-2809" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2810" class="MJXp-mi MJXp-italic">K</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2811" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2812" class="MJXp-mo">,</span><span id="MJXp-Span-2813" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2814" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2815" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2816" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-439-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3339" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3340" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3341" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3342" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3343" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3344" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3345" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3346" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3347" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3348" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3349" class="msubsup"><span style="display: inline-block; position: relative; width: 1.151em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.77em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3350" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3351" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3352" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3353" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3354" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3355" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3356" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3357" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3358" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3359" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3360" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3361" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3362" class="msubsup"><span style="display: inline-block; position: relative; width: 1.151em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.77em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3363" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3364" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3365" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-439" type="math/tex">e(G,K_1,H_1)=e(G,K_2,H_2)</script></span>.</p>
-
-<p>The advantage of strong Shoda pairs over Shoda pairs is that one can describe the simple algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2817" class="MJXp-math"><span id="MJXp-Span-2818" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-2819" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-2820" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2821" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2822" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2823" class="MJXp-mo">(</span><span id="MJXp-Span-2824" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2825" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-440-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3366" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3367" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3368" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3369" class="mi">G</span><span id="MathJax-Span-3370" class="msubsup"><span style="display: inline-block; position: relative; width: 0.975em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3371" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3372" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3373" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3374" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3375" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-440" type="math/tex">FGe_F(\chi)</script></span> as a matrix algebra of a <em>cyclotomic algebra</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>, see <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBORS">[ORS04]</a> for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2826" class="MJXp-math"><span id="MJXp-Span-2827" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2828" class="MJXp-mo">=</span><span id="MJXp-Span-2829" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2830" class="MJXp-mo">ℚ</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-441-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3376" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3377" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3378" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3379" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-3380" class="texatom"><span id="MathJax-Span-3381" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3382" class="mo">ℚ</span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-441" type="math/tex">F=ℚ </script></span> and <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBO">[Olt07]</a> for the general case).</p>
-
-<p>More precisely, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2831" class="MJXp-math"><span id="MJXp-Span-2832" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2833" class="MJXp-mo">ℚ</span></span><span id="MJXp-Span-2834" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-2835" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2836" class="MJXp-mo">(</span><span id="MJXp-Span-2837" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2838" class="MJXp-mo">,</span><span id="MJXp-Span-2839" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2840" class="MJXp-mo">,</span><span id="MJXp-Span-2841" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2842" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-442-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3383" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3384" class="mrow"><span id="MathJax-Span-3385" class="texatom"><span id="MathJax-Span-3386" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3387" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3388" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3389" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3390" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3391" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3392" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3393" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3394" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3395" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3396" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-442" type="math/tex">ℚ Ge(G,K,H)</script></span> is isomorphic to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2843" class="MJXp-math"><span id="MJXp-Span-2844" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2845" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-2846" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2847" class="MJXp-mo">(</span><span id="MJXp-Span-2848" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2849" class="MJXp-mo">ℚ</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2850" class="MJXp-mo">(</span><span id="MJXp-Span-2851" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2852" class="MJXp-mo">)</span><span id="MJXp-Span-2853" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-2854" class="MJXp-mo">∗</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-2856" class="MJXp-mi MJXp-italic">t</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-2855" class="MJXp-mi MJXp-italic">a</span></span></span></span></span></span><span id="MJXp-Span-2857" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-2858" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2859" class="MJXp-mo">/</span></span><span id="MJXp-Span-2860" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2861" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-443-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3397" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3398" class="mrow"><span id="MathJax-Span-3399" class="msubsup"><span style="display: inline-block; position: relative; width: 1.262em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3400" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3401" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3402" class="mo">(</span><span id="MathJax-Span-3403" class="texatom"><span id="MathJax-Span-3404" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3405" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3406" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3407" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3408" class="mo">)</span><span style="padding-left: 0.25em;" id="MathJax-Span-3409" class="msubsup"><span style="display: inline-block; position: relative; width: 0.952em; height: 0px;"><span style="position: absolute; clip: rect(3.375em, 1000.46em, 4.102em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-3410" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.46em, 1000.28em, 4.142em, -1000em); top: -4.343em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3411" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.013em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.142em, -1000em); top: -3.843em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3412" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3413" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3414" class="texatom"><span id="MathJax-Span-3415" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3416" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3417" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3418" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-443" type="math/tex">M_n(ℚ (\xi)*_a^t N/K)</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2862" class="MJXp-math"><span id="MJXp-Span-2863" class="MJXp-mi MJXp-italic">ξ</span></span></span><span style="" tabindex="0" id="MathJax-Element-444-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3419" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3420" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3421" class="mi">ξ</span></span></span></span></span></nobr></span><script id="MathJax-Element-444" type="math/tex">\xi</script></span> is a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2864" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2865" class="MJXp-mo">[</span><span id="MJXp-Span-2866" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-2867" class="MJXp-mo">:</span><span id="MJXp-Span-2868" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2869" class="MJXp-mo">]</span></span></span><span style="" tabindex="0" id="MathJax-Element-445-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3422" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3423" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3424" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3425" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3426" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3427" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3428" class="mo">]</span></span></span></span></span></nobr></span><script id="MathJax-Element-445" type="math/tex">[K:H]</script></span>-th root of unity, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2870" class="MJXp-math"><span id="MJXp-Span-2871" class="MJXp-mi MJXp-italic">N</span></span></span><span style="" tabindex="0" id="MathJax-Element-446-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3429" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3430" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3431" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-446" type="math/tex">N</script></span> is the normalizer of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2872" class="MJXp-math"><span id="MJXp-Span-2873" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-447-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3432" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3433" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3434" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-447" type="math/tex">H</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2874" class="MJXp-math"><span id="MJXp-Span-2875" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-448-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3435" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3436" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3437" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-448" type="math/tex">G</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2876" class="MJXp-math"><span id="MJXp-Span-2877" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2878" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2879" class="MJXp-mo">[</span><span id="MJXp-Span-2880" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-2881" class="MJXp-mo">:</span><span id="MJXp-Span-2882" class="MJXp-mi MJXp-italic">N</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2883" class="MJXp-mo">]</span></span></span><span style="" tabindex="0" id="MathJax-Element-449-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3438" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3439" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3440" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3441" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3442" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3443" class="mi">G</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3444" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3445" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3446" class="mo">]</span></span></span></span></span></nobr></span><script id="MathJax-Element-449" type="math/tex">n=[G:N]</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2884" class="MJXp-math"><span id="MJXp-Span-2885" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2886" class="MJXp-mo">ℚ</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2887" class="MJXp-mo">(</span><span id="MJXp-Span-2888" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2889" class="MJXp-mo">)</span><span id="MJXp-Span-2890" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-2891" class="MJXp-mo">∗</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-2893" class="MJXp-mi MJXp-italic">t</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-2892" class="MJXp-mi MJXp-italic">a</span></span></span></span></span></span><span id="MJXp-Span-2894" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-2895" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2896" class="MJXp-mo">/</span></span><span id="MJXp-Span-2897" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-450-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3447" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3448" class="mrow"><span id="MathJax-Span-3449" class="texatom"><span id="MathJax-Span-3450" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3451" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3452" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3453" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3454" class="mo">)</span><span style="padding-left: 0.25em;" id="MathJax-Span-3455" class="msubsup"><span style="display: inline-block; position: relative; width: 0.952em; height: 0px;"><span style="position: absolute; clip: rect(3.375em, 1000.46em, 4.102em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-3456" class="mo">∗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.46em, 1000.28em, 4.142em, -1000em); top: -4.343em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3457" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.013em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.43em, 4.142em, -1000em); top: -3.843em; left: 0.523em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3458" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3459" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3460" class="texatom"><span id="MathJax-Span-3461" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3462" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3463" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-450" type="math/tex">ℚ (\xi)*_a^t N/K</script></span> is a <em>crossed product</em> (see <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7FB21779832CE1CB"><span class="RefLink">9.6</span></a>) with action <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2898" class="MJXp-math"><span id="MJXp-Span-2899" class="MJXp-mi MJXp-italic">a</span></span></span><span style="" tabindex="0" id="MathJax-Element-451-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3464" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3465" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3466" class="mi">a</span></span></span></span></span></nobr></span><script id="MathJax-Element-451" type="math/tex">a</script></span> and twisting <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2900" class="MJXp-math"><span id="MJXp-Span-2901" class="MJXp-mi MJXp-italic">t</span></span></span><span style="" tabindex="0" id="MathJax-Element-452-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3467" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3468" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3469" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-452" type="math/tex">t</script></span> given as follows:</p>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2902" class="MJXp-math"><span id="MJXp-Span-2903" class="MJXp-mi MJXp-italic">x</span></span></span><span style="" tabindex="0" id="MathJax-Element-453-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3470" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3471" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3472" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-453" type="math/tex">x</script></span> be a fixed generator of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2904" class="MJXp-math"><span id="MJXp-Span-2905" class="MJXp-mi MJXp-italic">K</span><span id="MJXp-Span-2906" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2907" class="MJXp-mo">/</span></span><span id="MJXp-Span-2908" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-454-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3473" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3474" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3475" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span id="MathJax-Span-3476" class="texatom"><span id="MathJax-Span-3477" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3478" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3479" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-454" type="math/tex">K/H</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2909" class="MJXp-math"><span id="MJXp-Span-2910" class="MJXp-mi MJXp-italic">φ</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-2911" class="MJXp-mo">:</span><span id="MJXp-Span-2912" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-2913" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2914" class="MJXp-mo">/</span></span><span id="MJXp-Span-2915" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2916" class="MJXp-mo">→</span><span id="MJXp-Span-2917" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-2918" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2919" class="MJXp-mo">/</span></span><span id="MJXp-Span-2920" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-455-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3480" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3481" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3482" class="mi">φ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3483" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3484" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3485" class="texatom"><span id="MathJax-Span-3486" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3487" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3488" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3489" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3490" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3491" class="texatom"><span id="MathJax-Span-3492" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3493" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3494" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-455" type="math/tex">\varphi : N/K \rightarrow N/H</script></span> a fixed left inverse of the canonical projection <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2921" class="MJXp-math"><span id="MJXp-Span-2922" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-2923" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2924" class="MJXp-mo">/</span></span><span id="MJXp-Span-2925" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2926" class="MJXp-mo">→</span><span id="MJXp-Span-2927" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-2928" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-2929" class="MJXp-mo">/</span></span><span id="MJXp-Span-2930" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-456-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3495" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3496" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3497" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3498" class="texatom"><span id="MathJax-Span-3499" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3500" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3501" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3502" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3503" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3504" class="texatom"><span id="MathJax-Span-3505" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3506" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3507" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-456" type="math/tex">N/H\rightarrow N/K</script></span>. Then</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2931" class="MJXp-math MJXp-display"><span id="MJXp-Span-2932" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2933" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2934" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2935" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-2936" class="MJXp-mo">(</span><span id="MJXp-Span-2937" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-2938" class="MJXp-mo">)</span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2939" class="MJXp-mo">=</span><span id="MJXp-Span-2940" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2941" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2942" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2943" class="MJXp-mo">,</span><span id="MJXp-Span-2944" class="MJXp-mstyle"><span id="MJXp-Span-2945" class="MJXp-mtext"> if </span></span><span id="MJXp-Span-2946" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2947" class="MJXp-mi MJXp-italic">x</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2948" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2949" class="MJXp-mi MJXp-italic">φ</span><span id="MJXp-Span-2950" class="MJXp-mo">(</span><span id="MJXp-Span-2951" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-2952" class="MJXp-mo">)</span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2953" class="MJXp-mo">=</span><span id="MJXp-Span-2954" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2955" class="MJXp-mi MJXp-italic">x</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2956" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-457-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-3508" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3509" class="mrow"><span id="MathJax-Span-3510" class="msubsup"><span style="display: inline-block; position: relative; width: 1.646em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3511" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span id="MathJax-Span-3512" class="texatom"><span id="MathJax-Span-3513" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3514" class="mi">a</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3515" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3516" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3517" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3518" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-3519" class="msubsup"><span style="display: inline-block; position: relative; width: 0.726em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3520" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3521" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3522" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3523" class="mstyle"><span id="MathJax-Span-3524" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3525" class="mtext"> if </span></span></span><span id="MathJax-Span-3526" class="msubsup"><span style="display: inline-block; position: relative; width: 1.775em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3527" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.498em;"><span id="MathJax-Span-3528" class="texatom"><span id="MathJax-Span-3529" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3530" class="mi">φ</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3531" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3532" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3533" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3534" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-3535" class="msubsup"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3536" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.498em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3537" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span></div><script id="MathJax-Element-457" type="math/tex; mode=display">
- \xi^{a(r)} = \xi^i, \mbox{ if } x^{\varphi(r)}= x^i
- </script></p>
-
-<p>and</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2957" class="MJXp-math MJXp-display"><span id="MJXp-Span-2958" class="MJXp-mi MJXp-italic">t</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2959" class="MJXp-mo">(</span><span id="MJXp-Span-2960" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2961" class="MJXp-mo">,</span><span id="MJXp-Span-2962" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2963" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2964" class="MJXp-mo">=</span><span id="MJXp-Span-2965" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2966" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2967" class="MJXp-mi MJXp-italic MJXp-script">j</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2968" class="MJXp-mo">,</span><span id="MJXp-Span-2969" class="MJXp-mstyle"><span id="MJXp-Span-2970" class="MJXp-mtext"> if </span></span><span id="MJXp-Span-2971" class="MJXp-mi MJXp-italic">φ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2972" class="MJXp-mo">(</span><span id="MJXp-Span-2973" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-2974" class="MJXp-mi MJXp-italic">s</span><span id="MJXp-Span-2975" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-2976" class="MJXp-mo">)</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2977" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-2978" class="MJXp-mo">−</span><span id="MJXp-Span-2979" class="MJXp-mn">1</span></span></span><span id="MJXp-Span-2980" class="MJXp-mi MJXp-italic">φ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2981" class="MJXp-mo">(</span><span id="MJXp-Span-2982" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2983" class="MJXp-mo">)</span><span id="MJXp-Span-2984" class="MJXp-mi MJXp-italic">φ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2985" class="MJXp-mo">(</span><span id="MJXp-Span-2986" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-2987" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2988" class="MJXp-mo">=</span><span id="MJXp-Span-2989" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-2990" class="MJXp-mi MJXp-italic">x</span><span style="vertical-align: 0.5em;" id="MJXp-Span-2991" class="MJXp-mi MJXp-italic MJXp-script">j</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2992" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-458-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-3538" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3539" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3540" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3541" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3542" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3543" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3544" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3545" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3546" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-3547" class="msubsup"><span style="display: inline-block; position: relative; width: 0.726em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3548" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3549" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3550" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3551" class="mstyle"><span id="MathJax-Span-3552" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3553" class="mtext"> if </span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3554" class="mi">φ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3555" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3556" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3557" class="mi">s</span><span id="MathJax-Span-3558" class="msubsup"><span style="display: inline-block; position: relative; width: 1.246em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-3559" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.333em;"><span id="MathJax-Span-3560" class="texatom"><span id="MathJax-Span-3561" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3562" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3563" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3564" class="mi">φ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3565" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3566" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3567" class="mo">)</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3568" class="mi">φ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3569" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3570" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3571" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3572" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-3573" class="msubsup"><span style="display: inline-block; position: relative; width: 0.77em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3574" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.498em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3575" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3576" class="mo">,</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-458" type="math/tex; mode=display">
- t(r,s) = \xi^j, \mbox{ if } \varphi(rs)^{-1} \varphi(r)\varphi(s) = x^j,
- </script></p>
-
-<p>for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-2993" class="MJXp-math"><span id="MJXp-Span-2994" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-2995" class="MJXp-mo">,</span><span id="MJXp-Span-2996" class="MJXp-mi MJXp-italic">s</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-2997" class="MJXp-mo">∈</span><span id="MJXp-Span-2998" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-2999" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3000" class="MJXp-mo">/</span></span><span id="MJXp-Span-3001" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-459-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3577" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3578" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3579" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3580" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3581" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3582" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3583" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3584" class="texatom"><span id="MathJax-Span-3585" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3586" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3587" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-459" type="math/tex">r,s \in N/K</script></span> and integers <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3002" class="MJXp-math"><span id="MJXp-Span-3003" class="MJXp-mi MJXp-italic">i</span></span></span><span style="" tabindex="0" id="MathJax-Element-460-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3588" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3589" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3590" class="mi">i</span></span></span></span></span></nobr></span><script id="MathJax-Element-460" type="math/tex">i</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3004" class="MJXp-math"><span id="MJXp-Span-3005" class="MJXp-mi MJXp-italic">j</span></span></span><span style="" tabindex="0" id="MathJax-Element-461-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3591" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3592" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3593" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-461" type="math/tex">j</script></span>, see <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBORS">[ORS04]</a>. Notice that the cocycle is the one given by the natural extension</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3006" class="MJXp-math MJXp-display"><span id="MJXp-Span-3007" class="MJXp-mn">1</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3008" class="MJXp-mo">→</span><span id="MJXp-Span-3009" class="MJXp-mi MJXp-italic">K</span><span id="MJXp-Span-3010" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3011" class="MJXp-mo">/</span></span><span id="MJXp-Span-3012" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3013" class="MJXp-mo">→</span><span id="MJXp-Span-3014" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-3015" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3016" class="MJXp-mo">/</span></span><span id="MJXp-Span-3017" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3018" class="MJXp-mo">→</span><span id="MJXp-Span-3019" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-3020" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3021" class="MJXp-mo">/</span></span><span id="MJXp-Span-3022" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3023" class="MJXp-mo">→</span><span id="MJXp-Span-3024" class="MJXp-mn">1</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-462-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-3594" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3595" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3596" class="mn">1</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3597" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3598" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span id="MathJax-Span-3599" class="texatom"><span id="MathJax-Span-3600" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3601" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3602" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3603" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3604" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3605" class="texatom"><span id="MathJax-Span-3606" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3607" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3608" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3609" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3610" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3611" class="texatom"><span id="MathJax-Span-3612" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3613" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3614" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3615" class="mo">→</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3616" class="mn">1</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-462" type="math/tex; mode=display">
- 1 \rightarrow K/H \rightarrow N/H \rightarrow N/K \rightarrow 1
- </script></p>
-
-<p>where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3025" class="MJXp-math"><span id="MJXp-Span-3026" class="MJXp-mi MJXp-italic">K</span><span id="MJXp-Span-3027" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3028" class="MJXp-mo">/</span></span><span id="MJXp-Span-3029" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-463-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3617" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3618" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3619" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span id="MathJax-Span-3620" class="texatom"><span id="MathJax-Span-3621" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3622" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3623" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-463" type="math/tex">K/H</script></span> is identified with the multiplicative group generated by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3030" class="MJXp-math"><span id="MJXp-Span-3031" class="MJXp-mi MJXp-italic">ξ</span></span></span><span style="" tabindex="0" id="MathJax-Element-464-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3624" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3625" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3626" class="mi">ξ</span></span></span></span></span></nobr></span><script id="MathJax-Element-464" type="math/tex">\xi</script></span>. Furthermore the centre of the algebra is <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3032" class="MJXp-math"><span id="MJXp-Span-3033" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3034" class="MJXp-mo">ℚ</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3035" class="MJXp-mo">(</span><span id="MJXp-Span-3036" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3037" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-465-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3627" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3628" class="mrow"><span id="MathJax-Span-3629" class="texatom"><span id="MathJax-Span-3630" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3631" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3632" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3633" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3634" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-465" type="math/tex">ℚ (\chi)</script></span>, the field of character values over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3038" class="MJXp-math"><span id="MJXp-Span-3039" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3040" class="MJXp-mo">ℚ</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-466-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3635" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3636" class="mrow"><span id="MathJax-Span-3637" class="texatom"><span id="MathJax-Span-3638" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3639" class="mo">ℚ</span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-466" type="math/tex">ℚ </script></span>, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3041" class="MJXp-math"><span id="MJXp-Span-3042" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-3043" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3044" class="MJXp-mo">/</span></span><span id="MJXp-Span-3045" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-467-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3640" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3641" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3642" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3643" class="texatom"><span id="MathJax-Span-3644" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3645" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3646" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-467" type="math/tex">N/K</script></span> is isomorphic to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3046" class="MJXp-math"><span id="MJXp-Span-3047" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-3048" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-3049" class="MJXp-mi MJXp-italic">l</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3050" class="MJXp-mo">(</span><span id="MJXp-Span-3051" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3052" class="MJXp-mo">ℚ</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3053" class="MJXp-mo">(</span><span id="MJXp-Span-3054" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3055" class="MJXp-mo">)</span><span id="MJXp-Span-3056" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3057" class="MJXp-mo">/</span></span><span id="MJXp-Span-3058" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3059" class="MJXp-mo">ℚ</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3060" class="MJXp-mo">(</span><span id="MJXp-Span-3061" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3062" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3063" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-468-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3647" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3648" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3649" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3650" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3651" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3652" class="mo">(</span><span id="MathJax-Span-3653" class="texatom"><span id="MathJax-Span-3654" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3655" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3656" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3657" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3658" class="mo">)</span><span id="MathJax-Span-3659" class="texatom"><span id="MathJax-Span-3660" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3661" class="mo">/</span></span></span><span id="MathJax-Span-3662" class="texatom"><span id="MathJax-Span-3663" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3664" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3665" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3666" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3667" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3668" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-468" type="math/tex">Gal(ℚ (\xi)/ℚ (\chi))</script></span>.</p>
-
-<p>If the rational field is changed to an arbitrary ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3064" class="MJXp-math"><span id="MJXp-Span-3065" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-469-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3669" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3670" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3671" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-469" type="math/tex">F</script></span> of characteristic <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3066" class="MJXp-math"><span id="MJXp-Span-3067" class="MJXp-mn">0</span></span></span><span style="" tabindex="0" id="MathJax-Element-470-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3672" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3673" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3674" class="mn">0</span></span></span></span></span></nobr></span><script id="MathJax-Element-470" type="math/tex">0</script></span> then the Wedderburn component <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3068" class="MJXp-math"><span id="MJXp-Span-3069" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3070" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3071" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3072" class="MJXp-mo">(</span><span id="MJXp-Span-3073" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3074" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-471-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3675" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3676" class="mrow"><span id="MathJax-Span-3677" class="msubsup"><span style="display: inline-block; position: relative; width: 1.142em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3678" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3679" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3680" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3681" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3682" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-471" type="math/tex">A_F(\chi)</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3075" class="MJXp-math"><span id="MJXp-Span-3076" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3077" class="MJXp-mo">=</span><span id="MJXp-Span-3078" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3079" class="MJXp-mo">(</span><span id="MJXp-Span-3080" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3081" class="MJXp-mo">,</span><span id="MJXp-Span-3082" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3083" class="MJXp-mo">,</span><span id="MJXp-Span-3084" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3085" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-472-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3683" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3684" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3685" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3686" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3687" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3688" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3689" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3690" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3691" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3692" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3693" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3694" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-472" type="math/tex">\chi = \chi(G,K,H)</script></span> is isomorphic to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3086" class="MJXp-math"><span id="MJXp-Span-3087" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3088" class="MJXp-mo">(</span><span id="MJXp-Span-3089" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3090" class="MJXp-mo">)</span><span id="MJXp-Span-3091" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-3092" class="MJXp-mo">⊗</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3093" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-3094" class="MJXp-mrow"><span id="MJXp-Span-3095" class="MJXp-mo">ℚ</span></span><span id="MJXp-Span-3096" class="MJXp-mo">(</span><span id="MJXp-Span-3097" class="MJXp-mi MJXp-italic">χ</span><span id="MJXp-Span-3098" class="MJXp-mo">)</span></span></span><span id="MJXp-Span-3099" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3100" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3101" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-3102" class="MJXp-mo">ℚ</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3103" class="MJXp-mo">(</span><span id="MJXp-Span-3104" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3105" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-473-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3695" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3696" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3697" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3698" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3699" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3700" class="mo">)</span><span style="padding-left: 0.25em;" id="MathJax-Span-3701" class="msubsup"><span style="display: inline-block; position: relative; width: 2.281em; height: 0px;"><span style="position: absolute; clip: rect(3.223em, 1000.79em, 4.254em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-3702" class="mo">⊗</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.842em;"><span id="MathJax-Span-3703" class="texatom"><span id="MathJax-Span-3704" class="mrow"><span id="MathJax-Span-3705" class="texatom"><span id="MathJax-Span-3706" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3707" class="mo">ℚ</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3708" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3709" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.029em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3710" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.25em;" id="MathJax-Span-3711" class="msubsup"><span style="display: inline-block; position: relative; width: 1.227em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3712" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span id="MathJax-Span-3713" class="texatom"><span id="MathJax-Span-3714" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3715" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3716" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3717" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3718" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-473" type="math/tex">F(\chi)\otimes_{ℚ (\chi)}A_ℚ (\chi)</script></span>. Using the description given above of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3106" class="MJXp-math"><span id="MJXp-Span-3107" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3108" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3109" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-3110" class="MJXp-mo">ℚ</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3111" class="MJXp-mo">(</span><span id="MJXp-Span-3112" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3113" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3114" class="MJXp-mo">=</span><span id="MJXp-Span-3115" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3116" class="MJXp-mo">ℚ</span></span><span id="MJXp-Span-3117" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-3118" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3119" class="MJXp-mo">(</span><span id="MJXp-Span-3120" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3121" class="MJXp-mo">,</span><span id="MJXp-Span-3122" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3123" class="MJXp-mo">,</span><span id="MJXp-Span-3124" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3125" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-474-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3719" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3720" class="mrow"><span id="MathJax-Span-3721" class="msubsup"><span style="display: inline-block; position: relative; width: 1.227em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3722" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span id="MathJax-Span-3723" class="texatom"><span id="MathJax-Span-3724" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3725" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3726" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3727" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3728" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3729" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-3730" class="texatom"><span id="MathJax-Span-3731" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3732" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3733" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3734" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3735" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3736" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3737" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3738" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3739" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3740" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3741" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-474" type="math/tex">A_ℚ (\chi)=ℚ G e(G,K,H)</script></span> one can easily describe <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3126" class="MJXp-math"><span id="MJXp-Span-3127" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3128" class="MJXp-mi MJXp-italic">A</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3129" class="MJXp-mi MJXp-italic MJXp-script">F</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3130" class="MJXp-mo">(</span><span id="MJXp-Span-3131" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3132" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-475-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3742" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3743" class="mrow"><span id="MathJax-Span-3744" class="msubsup"><span style="display: inline-block; position: relative; width: 1.142em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3745" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3746" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3747" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3748" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3749" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-475" type="math/tex">A_F(\chi)</script></span> as <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3133" class="MJXp-math"><span id="MJXp-Span-3134" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3135" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3136" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-3137" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-3138" class="MJXp-mi MJXp-italic">d</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3139" class="MJXp-mo">(</span><span id="MJXp-Span-3140" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3141" class="MJXp-mo">(</span><span id="MJXp-Span-3142" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3143" class="MJXp-mo">)</span><span id="MJXp-Span-3144" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3145" class="MJXp-mo">/</span></span><span id="MJXp-Span-3146" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3147" class="MJXp-mo">(</span><span id="MJXp-Span-3148" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3149" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3150" class="MJXp-mo">,</span><span id="MJXp-Span-3151" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-3152" class="MJXp-mi MJXp-italic">t</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3153" class="MJXp-mo MJXp-script">′</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3154" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-476-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3750" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3751" class="mrow"><span id="MathJax-Span-3752" class="msubsup"><span style="display: inline-block; position: relative; width: 1.634em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.87em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3753" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.833em;"><span id="MathJax-Span-3754" class="texatom"><span id="MathJax-Span-3755" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3756" class="mi">n</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3757" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3758" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3759" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3760" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3761" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3762" class="mo">)</span><span id="MathJax-Span-3763" class="texatom"><span id="MathJax-Span-3764" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3765" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3766" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3767" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3768" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3769" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3770" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3771" class="msup"><span style="display: inline-block; position: relative; width: 0.667em; height: 0px;"><span style="position: absolute; clip: rect(3.3em, 1000.3em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3772" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.351em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-3773" class="mo">′</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3774" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-476" type="math/tex">M_{nd}(F(\xi)/F(\chi),t')</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3155" class="MJXp-math"><span id="MJXp-Span-3156" class="MJXp-mi MJXp-italic">d</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3157" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3158" class="MJXp-mo">[</span><span id="MJXp-Span-3159" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3160" class="MJXp-mo">ℚ</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3161" class="MJXp-mo">(</span><span id="MJXp-Span-3162" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3163" class="MJXp-mo">)</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-3164" class="MJXp-mo">:</span><span id="MJXp-Span-3165" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3166" class="MJXp-mo">ℚ</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3167" class="MJXp-mo">(</span><span id="MJXp-Span-3168" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3169" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3170" class="MJXp-mo">]</span><span id="MJXp-Span-3171" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3172" class="MJXp-mo">/</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3173" class="MJXp-mo">[</span><span id="MJXp-Span-3174" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3175" class="MJXp-mo">(</span><span id="MJXp-Span-3176" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3177" class="MJXp-mo">)</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-3178" class="MJXp-mo">:</span><span id="MJXp-Span-3179" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3180" class="MJXp-mo">(</span><span id="MJXp-Span-3181" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3182" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3183" class="MJXp-mo">]</span></span></span><span style="" tabindex="0" id="MathJax-Element-477-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3775" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3776" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3777" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3778" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3779" class="mo">[</span><span id="MathJax-Span-3780" class="texatom"><span id="MathJax-Span-3781" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3782" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3783" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3784" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3785" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3786" class="mo">:</span><span style="padding-left: 0.313em;" id="MathJax-Span-3787" class="texatom"><span id="MathJax-Span-3788" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3789" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3790" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3791" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3792" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3793" class="mo">]</span><span id="MathJax-Span-3794" class="texatom"><span id="MathJax-Span-3795" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3796" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3797" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3798" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3799" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3800" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3801" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3802" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3803" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3804" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3805" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3806" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3807" class="mo">]</span></span></span></span></span></nobr></span><script id="MathJax-Element-477" type="math/tex">d=[ℚ (\xi): ℚ(\chi)]/[F(\xi):F(\chi)]</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3184" class="MJXp-math"><span id="MJXp-Span-3185" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-3186" class="MJXp-mi MJXp-italic">t</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3187" class="MJXp-mo MJXp-script">′</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-478-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3808" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3809" class="mrow"><span id="MathJax-Span-3810" class="msup"><span style="display: inline-block; position: relative; width: 0.667em; height: 0px;"><span style="position: absolute; clip: rect(3.3em, 1000.3em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3811" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.351em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-3812" class="mo">′</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-478" type="math/tex">t'</script></span> is the restriction to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3188" class="MJXp-math"><span id="MJXp-Span-3189" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-3190" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-3191" class="MJXp-mi MJXp-italic">l</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3192" class="MJXp-mo">(</span><span id="MJXp-Span-3193" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3194" class="MJXp-mo">(</span><span id="MJXp-Span-3195" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3196" class="MJXp-mo">)</span><span id="MJXp-Span-3197" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3198" class="MJXp-mo">/</span></span><span id="MJXp-Span-3199" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3200" class="MJXp-mo">(</span><span id="MJXp-Span-3201" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3202" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3203" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-479-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3813" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3814" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3815" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3816" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3817" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3818" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3819" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3820" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3821" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3822" class="mo">)</span><span id="MathJax-Span-3823" class="texatom"><span id="MathJax-Span-3824" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3825" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3826" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3827" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3828" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3829" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3830" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-479" type="math/tex">Gal(F(\xi)/F(\chi))</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3204" class="MJXp-math"><span id="MJXp-Span-3205" class="MJXp-mi MJXp-italic">t</span></span></span><span style="" tabindex="0" id="MathJax-Element-480-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3831" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3832" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3833" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-480" type="math/tex">t</script></span> (a cocycle of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3206" class="MJXp-math"><span id="MJXp-Span-3207" class="MJXp-mi MJXp-italic">N</span><span id="MJXp-Span-3208" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3209" class="MJXp-mo">/</span></span><span id="MJXp-Span-3210" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3211" class="MJXp-mo">=</span><span id="MJXp-Span-3212" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-3213" class="MJXp-mi MJXp-italic">a</span><span id="MJXp-Span-3214" class="MJXp-mi MJXp-italic">l</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3215" class="MJXp-mo">(</span><span id="MJXp-Span-3216" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3217" class="MJXp-mo">ℚ</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3218" class="MJXp-mo">(</span><span id="MJXp-Span-3219" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3220" class="MJXp-mo">)</span><span id="MJXp-Span-3221" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3222" class="MJXp-mo">/</span></span><span id="MJXp-Span-3223" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3224" class="MJXp-mo">ℚ</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3225" class="MJXp-mo">(</span><span id="MJXp-Span-3226" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3227" class="MJXp-mo">)</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3228" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-481-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3834" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3835" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3836" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span id="MathJax-Span-3837" class="texatom"><span id="MathJax-Span-3838" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3839" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3840" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3841" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3842" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3843" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3844" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3845" class="mo">(</span><span id="MathJax-Span-3846" class="texatom"><span id="MathJax-Span-3847" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3848" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3849" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3850" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3851" class="mo">)</span><span id="MathJax-Span-3852" class="texatom"><span id="MathJax-Span-3853" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3854" class="mo">/</span></span></span><span id="MathJax-Span-3855" class="texatom"><span id="MathJax-Span-3856" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3857" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3858" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3859" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3860" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3861" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-481" type="math/tex">N/K = Gal(ℚ (\xi)/ℚ (\chi))</script></span>).</p>
-
-<p><a id="X84C694978557EFE5" name="X84C694978557EFE5"></a></p>
-
-<h4>9.16 <span class="Heading">Strongly monomial characters and strongly monomial groups</span></h4>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3229" class="MJXp-math"><span id="MJXp-Span-3230" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-482-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3862" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3863" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3864" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-482" type="math/tex">G</script></span> be a finite group an <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3231" class="MJXp-math"><span id="MJXp-Span-3232" class="MJXp-mi MJXp-italic">χ</span></span></span><span style="" tabindex="0" id="MathJax-Element-483-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3865" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3866" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3867" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-483" type="math/tex">\chi</script></span> an irreducible character of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3233" class="MJXp-math"><span id="MJXp-Span-3234" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-484-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3868" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3869" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3870" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-484" type="math/tex">G</script></span>.</p>
-
-<p>One says that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3235" class="MJXp-math"><span id="MJXp-Span-3236" class="MJXp-mi MJXp-italic">χ</span></span></span><span style="" tabindex="0" id="MathJax-Element-485-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3871" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3872" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3873" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-485" type="math/tex">\chi</script></span> is <em>strongly monomial</em> if there is a <em>strong Shoda pair</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3237" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3238" class="MJXp-mo">(</span><span id="MJXp-Span-3239" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3240" class="MJXp-mo">,</span><span id="MJXp-Span-3241" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3242" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-486-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3874" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3875" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3876" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3877" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3878" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3879" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3880" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-486" type="math/tex">(K,H)</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3243" class="MJXp-math"><span id="MJXp-Span-3244" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-487-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3881" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3882" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3883" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-487" type="math/tex">G</script></span> and a linear character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3245" class="MJXp-math"><span id="MJXp-Span-3246" class="MJXp-mi MJXp-italic">θ</span></span></span><span style="" tabindex="0" id="MathJax-Element-488-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3884" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3885" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3886" class="mi">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-488" type="math/tex">\theta</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3247" class="MJXp-math"><span id="MJXp-Span-3248" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-489-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3887" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3888" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3889" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-489" type="math/tex">K</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3249" class="MJXp-math"><span id="MJXp-Span-3250" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-490-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3890" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3891" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3892" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-490" type="math/tex">G</script></span> with kernel <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3251" class="MJXp-math"><span id="MJXp-Span-3252" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-491-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3893" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3894" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3895" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-491" type="math/tex">H</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3253" class="MJXp-math"><span id="MJXp-Span-3254" class="MJXp-mi MJXp-italic">χ</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3255" class="MJXp-mo">=</span><span id="MJXp-Span-3256" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3257" class="MJXp-mi MJXp-italic">θ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3258" class="MJXp-mi MJXp-italic MJXp-script">G</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-492-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3896" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3897" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3898" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3899" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-3900" class="msubsup"><span style="display: inline-block; position: relative; width: 1.134em; height: 0px;"><span style="position: absolute; clip: rect(3.168em, 1000.49em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3901" class="mi">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.548em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3902" class="mi">G</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-492" type="math/tex">\chi=\theta^G</script></span>.</p>
-
-<p>The group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3259" class="MJXp-math"><span id="MJXp-Span-3260" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-493-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3903" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3904" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3905" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-493" type="math/tex">G</script></span> is <em>strongly monomial</em> if every irreducible character of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3261" class="MJXp-math"><span id="MJXp-Span-3262" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-494-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3906" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3907" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3908" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-494" type="math/tex">G</script></span> is strongly monomial.</p>
-
-<p>Strong Shoda pairs where firstly introduced by Olivieri, del RÃo and
-Simón who proved that every abelian-by-supersolvable group is strongly
-monomial <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBORS">[ORS04]</a>.
- The algorithm to compute the Wedderburn decomposition of rational group
- algebras for strongly monomial groups was explained in <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBOR">[OR03]</a>. This method was extended for semisimple finite group algebras by Broche Cristo and del RÃo in <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBBR">[BR07]</a> (see Section <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.17</span></a>). Finally, Olteanu <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBO">[Olt07]</a> shows how to compute the <em>Wedderburn decomposition</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3263" class="MJXp-math"><span id="MJXp-Span-3264" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-495-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3909" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3910" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3911" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-495" type="math/tex">G</script></span> but also the strong Shoda pairs of the subgroups of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3265" class="MJXp-math"><span id="MJXp-Span-3266" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-496-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3912" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3913" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3914" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-496" type="math/tex">G</script></span>.</p>
-
-<p>.</p>
-
-<p><a id="X800D8C5087D79DC8" name="X800D8C5087D79DC8"></a></p>
-
-<h4>9.17 <span class="Heading">Cyclotomic Classes and Strong Shoda Pairs</span></h4>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3267" class="MJXp-math"><span id="MJXp-Span-3268" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-497-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3915" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3916" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3917" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-497" type="math/tex">G</script></span> be a finite group and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3269" class="MJXp-math"><span id="MJXp-Span-3270" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-498-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3918" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3919" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3920" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-498" type="math/tex">F</script></span> a finite field of order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3271" class="MJXp-math"><span id="MJXp-Span-3272" class="MJXp-mi MJXp-italic">q</span></span></span><span style="" tabindex="0" id="MathJax-Element-499-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3921" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3922" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3923" class="mi">q</span></span></span></span></span></nobr></span><script id="MathJax-Element-499" type="math/tex">q</script></span>, coprime to the order of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3273" class="MJXp-math"><span id="MJXp-Span-3274" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-500-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3924" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3925" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3926" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-500" type="math/tex">G</script></span>.</p>
-
-<p>Given a positive integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3275" class="MJXp-math"><span id="MJXp-Span-3276" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-501-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3927" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3928" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3929" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-501" type="math/tex">n</script></span>, coprime to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3277" class="MJXp-math"><span id="MJXp-Span-3278" class="MJXp-mi MJXp-italic">q</span></span></span><span style="" tabindex="0" id="MathJax-Element-502-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3930" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3931" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3932" class="mi">q</span></span></span></span></span></nobr></span><script id="MathJax-Element-502" type="math/tex">q</script></span>, the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3279" class="MJXp-math"><span id="MJXp-Span-3280" class="MJXp-mi MJXp-italic">q</span></span></span><span style="" tabindex="0" id="MathJax-Element-503-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3933" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3934" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3935" class="mi">q</span></span></span></span></span></nobr></span><script id="MathJax-Element-503" type="math/tex">q</script></span>-<em>cyclotomic classes</em> modulo <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3281" class="MJXp-math"><span id="MJXp-Span-3282" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-504-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3936" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3937" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3938" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-504" type="math/tex">n</script></span> are the set of residue classes module <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3283" class="MJXp-math"><span id="MJXp-Span-3284" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-505-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3939" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3940" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3941" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-505" type="math/tex">n</script></span> of the form</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3285" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3286" class="MJXp-mo">{</span><span id="MJXp-Span-3287" class="MJXp-mi MJXp-italic">i</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3288" class="MJXp-mo">,</span><span id="MJXp-Span-3289" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-3290" class="MJXp-mi MJXp-italic">q</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3291" class="MJXp-mo">,</span><span id="MJXp-Span-3292" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-3293" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3294" class="MJXp-mi MJXp-italic">q</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3295" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3296" class="MJXp-mo">,</span><span id="MJXp-Span-3297" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-3298" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3299" class="MJXp-mi MJXp-italic">q</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3300" class="MJXp-mn MJXp-script">3</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3301" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3302" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3303" class="MJXp-mo">}</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-506-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-3942" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3943" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3944" class="mo">{</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3945" class="mi">i</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3946" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3947" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3948" class="mi">q</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3949" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3950" class="mi">i</span><span id="MathJax-Span-3951" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3952" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3953" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3954" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3955" class="mi">i</span><span id="MathJax-Span-3956" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3957" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3958" class="mn">3</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3959" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3960" class="mo">…</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3961" class="mo">}</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-506" type="math/tex; mode=display">
- \{i,iq,iq^2,iq^3, \dots \}
- </script></p>
-
-<p>The <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3304" class="MJXp-math"><span id="MJXp-Span-3305" class="MJXp-mi MJXp-italic">q</span></span></span><span style="" tabindex="0" id="MathJax-Element-507-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3962" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3963" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3964" class="mi">q</span></span></span></span></span></nobr></span><script id="MathJax-Element-507" type="math/tex">q</script></span>-cyclotomic classes module <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3306" class="MJXp-math"><span id="MJXp-Span-3307" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-508-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3965" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3966" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3967" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-508" type="math/tex">n</script></span> form a partition of the set of residue classes module <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3308" class="MJXp-math"><span id="MJXp-Span-3309" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-509-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3968" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3969" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3970" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-509" type="math/tex">n</script></span>.</p>
-
-<p>A <em>generating cyclotomic class </em> module <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3310" class="MJXp-math"><span id="MJXp-Span-3311" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-510-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3971" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3972" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3973" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-510" type="math/tex">n</script></span> is a cyclotomic class containing a generator of the additive group of residue classes module <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3312" class="MJXp-math"><span id="MJXp-Span-3313" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-511-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3974" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3975" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3976" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-511" type="math/tex">n</script></span>, or equivalently formed by integers coprime to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3314" class="MJXp-math"><span id="MJXp-Span-3315" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-512-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3977" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3978" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3979" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-512" type="math/tex">n</script></span>.</p>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3316" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3317" class="MJXp-mo">(</span><span id="MJXp-Span-3318" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3319" class="MJXp-mo">,</span><span id="MJXp-Span-3320" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3321" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-513-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3980" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3981" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3982" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3983" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3984" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3985" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3986" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-513" type="math/tex">(K,H)</script></span> be a strong Shoda pair (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3322" class="MJXp-math"><span id="MJXp-Span-3323" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-514-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3987" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3988" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3989" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-514" type="math/tex">G</script></span> and set <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3324" class="MJXp-math"><span id="MJXp-Span-3325" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3326" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3327" class="MJXp-mo">[</span><span id="MJXp-Span-3328" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-3329" class="MJXp-mo">:</span><span id="MJXp-Span-3330" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3331" class="MJXp-mo">]</span></span></span><span style="" tabindex="0" id="MathJax-Element-515-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3990" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-3991" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3992" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3993" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3994" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3995" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3996" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3997" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3998" class="mo">]</span></span></span></span></span></nobr></span><script id="MathJax-Element-515" type="math/tex">n=[K:H]</script></span>. Fix a primitive <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3332" class="MJXp-math"><span id="MJXp-Span-3333" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-516-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-3999" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4000" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4001" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-516" type="math/tex">n</script></span>-th root of unity <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3334" class="MJXp-math"><span id="MJXp-Span-3335" class="MJXp-mi MJXp-italic">ξ</span></span></span><span style="" tabindex="0" id="MathJax-Element-517-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4002" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4003" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4004" class="mi">ξ</span></span></span></span></span></nobr></span><script id="MathJax-Element-517" type="math/tex">\xi</script></span> in some extension of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3336" class="MJXp-math"><span id="MJXp-Span-3337" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-518-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4005" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4006" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4007" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-518" type="math/tex">F</script></span> and an element <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3338" class="MJXp-math"><span id="MJXp-Span-3339" class="MJXp-mi MJXp-italic">g</span></span></span><span style="" tabindex="0" id="MathJax-Element-519-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4008" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4009" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4010" class="mi">g</span></span></span></span></span></nobr></span><script id="MathJax-Element-519" type="math/tex">g</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3340" class="MJXp-math"><span id="MJXp-Span-3341" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-520-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4011" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4012" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4013" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-520" type="math/tex">K</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3342" class="MJXp-math"><span id="MJXp-Span-3343" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-3344" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-521-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4014" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4015" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4016" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4017" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-521" type="math/tex">gH</script></span> is a generator of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3345" class="MJXp-math"><span id="MJXp-Span-3346" class="MJXp-mi MJXp-italic">K</span><span id="MJXp-Span-3347" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3348" class="MJXp-mo">/</span></span><span id="MJXp-Span-3349" class="MJXp-mi MJXp-italic">H</span></span></span><span style="" tabindex="0" id="MathJax-Element-522-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4018" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4019" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4020" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span id="MathJax-Span-4021" class="texatom"><span id="MathJax-Span-4022" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4023" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4024" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-522" type="math/tex">K/H</script></span>. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3350" class="MJXp-math"><span id="MJXp-Span-3351" class="MJXp-mi MJXp-italic">C</span></span></span><span style="" tabindex="0" id="MathJax-Element-523-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4025" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4026" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4027" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-523" type="math/tex">C</script></span> be a generating <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3352" class="MJXp-math"><span id="MJXp-Span-3353" class="MJXp-mi MJXp-italic">q</span></span></span><span style="" tabindex="0" id="MathJax-Element-524-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4028" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4029" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4030" class="mi">q</span></span></span></span></span></nobr></span><script id="MathJax-Element-524" type="math/tex">q</script></span>-cyclotomic class modulo <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3354" class="MJXp-math"><span id="MJXp-Span-3355" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-525-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4031" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4032" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4033" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-525" type="math/tex">n</script></span>. Then set</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3356" class="MJXp-math MJXp-display"><span id="MJXp-Span-3357" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3358" class="MJXp-mi MJXp-italic">ε</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3359" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3360" class="MJXp-mo">(</span><span id="MJXp-Span-3361" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3362" class="MJXp-mo">,</span><span id="MJXp-Span-3363" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3364" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3365" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3366" class="MJXp-mo">[</span><span id="MJXp-Span-3367" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-3368" class="MJXp-mo">:</span><span id="MJXp-Span-3369" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-3370" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-3371" class="MJXp-mo">]</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3372" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-3373" class="MJXp-mo">−</span><span id="MJXp-Span-3374" class="MJXp-mn">1</span></span></span><span id="MJXp-Span-3375" class="MJXp-mrow"><span id="MJXp-Span-3376" class="MJXp-munderover"><span><span class="MJXp-over"><span style="margin-bottom: -0.92em;" class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-3378" class="MJXp-mo">ˆ</span></span><span class=""><span id="MJXp-Span-3377" class="MJXp-mi MJXp-italic">H</span></span></span></span></span></span><span id="MJXp-Span-3379" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-3385" class="MJXp-mrow"><span id="MJXp-Span-3386" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-3387" class="MJXp-mo">−</span><span id="MJXp-Span-3388" class="MJXp-mn">1</span></span></span><span class=""><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-3380" class="MJXp-mo"><span class="MJXp-largeop">∑</span></span></span></span></span><span class=" MJXp-script"><span style="margin-left: 0px;" id="MJXp-Span-3381" class="MJXp-mrow"><span id="MJXp-Span-3382" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-3383" class="MJXp-mo">=</span><span id="MJXp-Span-3384" class="MJXp-mn">0</span></span></span></span><span id="MJXp-Span-3389" class="MJXp-mi MJXp-italic">t</span><span id="MJXp-Span-3390" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3391" class="MJXp-mo">(</span><span id="MJXp-Span-3392" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3393" class="MJXp-mi MJXp-italic">ξ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3394" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-3395" class="MJXp-mo">−</span><span id="MJXp-Span-3396" class="MJXp-mi MJXp-italic">c</span><span id="MJXp-Span-3397" class="MJXp-mi MJXp-italic">i</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3398" class="MJXp-mo">)</span><span id="MJXp-Span-3399" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3400" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3401" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3402" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processed"><span style="" tabindex="0" id="MathJax-Element-526-Frame" class="MathJax"><nobr><span role="math" id="MathJax-Span-4034" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4035" class="mrow"><span id="MathJax-Span-4036" class="msubsup"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4037" class="mi">ε</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4038" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4039" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4040" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4041" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4042" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4043" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-4044" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-4045" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4046" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-4047" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-4048" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span id="MathJax-Span-4049" class="msubsup"><span style="display: inline-block; position: relative; width: 1.246em; height: 0px;"><span style="position: absolute; clip: rect(3.184em, 1000.24em, 4.291em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-4050" class="mo">]</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.333em;"><span id="MathJax-Span-4051" class="texatom"><span id="MathJax-Span-4052" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-4053" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-4054" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-4055" class="texatom"><span id="MathJax-Span-4056" class="mrow"><span id="MathJax-Span-4057" class="munderover"><span style="display: inline-block; position: relative; width: 1.035em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.77em, 4.135em, -1000em); top: -3.99em; left: 0.105em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4058" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.069em, 1000.98em, 3.571em, -1000em); top: -4.2em; left: 0.056em;"><span style="" id="MathJax-Span-4059" class="mo"><span style="font-family: STIXSizeTwoSym;">ˆ</span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-4060" class="munderover"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-4061" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.02em, 4.245em, -1000em); top: -2.84em; left: 0.129em;"><span id="MathJax-Span-4062" class="texatom"><span id="MathJax-Span-4063" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4064" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-4065" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-4066" class="mn">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.268em, 1001.12em, 4.141em, -1000em); top: -5.19em; left: 0.05em;"><span id="MathJax-Span-4067" class="texatom"><span id="MathJax-Span-4068" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4069" class="mi">n</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-4070" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-4071" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4072" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4073" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4074" class="mo">(</span><span id="MathJax-Span-4075" class="msubsup"><span style="display: inline-block; position: relative; width: 1.524em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.45em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4076" class="mi">ξ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.454em;"><span id="MathJax-Span-4077" class="texatom"><span id="MathJax-Span-4078" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-4079" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4080" class="mi">c</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4081" class="mi">i</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4082" class="mo">)</span><span id="MathJax-Span-4083" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4084" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4085" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4086" class="mo">,</span></span></span></span></span></nobr></span></div><script id="MathJax-Element-526" type="math/tex; mode=display">
- \varepsilon_C(K,H) = [K:H]^{-1} \widehat{H} \sum_{i=0}^{n-1} tr(\xi^{-ci})g^i,
- </script></p>
-
-<p>where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3403" class="MJXp-math"><span id="MJXp-Span-3404" class="MJXp-mi MJXp-italic">c</span></span></span><span style="" tabindex="0" id="MathJax-Element-527-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4087" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4088" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4089" class="mi">c</span></span></span></span></span></nobr></span><script id="MathJax-Element-527" type="math/tex">c</script></span> is an arbitrary element of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3405" class="MJXp-math"><span id="MJXp-Span-3406" class="MJXp-mi MJXp-italic">C</span></span></span><span style="" tabindex="0" id="MathJax-Element-528-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4090" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4091" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4092" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-528" type="math/tex">C</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3407" class="MJXp-math"><span id="MJXp-Span-3408" class="MJXp-mi MJXp-italic">t</span><span id="MJXp-Span-3409" class="MJXp-mi MJXp-italic">r</span></span></span><span style="" tabindex="0" id="MathJax-Element-529-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4093" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4094" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4095" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4096" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-529" type="math/tex">tr</script></span> is the trace map of the field extension <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3410" class="MJXp-math"><span id="MJXp-Span-3411" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3412" class="MJXp-mo">(</span><span id="MJXp-Span-3413" class="MJXp-mi MJXp-italic">ξ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3414" class="MJXp-mo">)</span><span id="MJXp-Span-3415" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3416" class="MJXp-mo">/</span></span><span id="MJXp-Span-3417" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-530-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4097" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4098" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4099" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4100" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4101" class="mi">ξ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4102" class="mo">)</span><span id="MathJax-Span-4103" class="texatom"><span id="MathJax-Span-4104" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4105" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4106" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-530" type="math/tex">F(\xi)/F</script></span>. Then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3418" class="MJXp-math"><span id="MJXp-Span-3419" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3420" class="MJXp-mi MJXp-italic">ε</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3421" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3422" class="MJXp-mo">(</span><span id="MJXp-Span-3423" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3424" class="MJXp-mo">,</span><span id="MJXp-Span-3425" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3426" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-531-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4107" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4108" class="mrow"><span id="MathJax-Span-4109" class="msubsup"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4110" class="mi">ε</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4111" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4112" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4113" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4114" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4115" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4116" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-531" type="math/tex">\varepsilon_C(K,H)</script></span> does not depend on the choice of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3427" class="MJXp-math"><span id="MJXp-Span-3428" class="MJXp-mi MJXp-italic">c</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3429" class="MJXp-mo">∈</span><span id="MJXp-Span-3430" class="MJXp-mi MJXp-italic">C</span></span></span><span style="" tabindex="0" id="MathJax-Element-532-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4117" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4118" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4119" class="mi">c</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-4120" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-4121" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-532" type="math/tex">c \in C</script></span> and is a <em>primitive central idempotent</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3431" class="MJXp-math"><span id="MJXp-Span-3432" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-3433" class="MJXp-mi MJXp-italic">K</span></span></span><span style="" tabindex="0" id="MathJax-Element-533-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4122" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4123" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4124" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4125" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-533" type="math/tex">FK</script></span>.</p>
-
-<p>Finally, let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3434" class="MJXp-math"><span id="MJXp-Span-3435" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3436" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3437" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3438" class="MJXp-mo">(</span><span id="MJXp-Span-3439" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3440" class="MJXp-mo">,</span><span id="MJXp-Span-3441" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3442" class="MJXp-mo">,</span><span id="MJXp-Span-3443" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3444" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-534-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4126" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4127" class="mrow"><span id="MathJax-Span-4128" class="msubsup"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4129" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4130" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4131" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4132" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4133" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4134" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4135" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4136" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4137" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-534" type="math/tex">e_C(G,K,H)</script></span> denote the sum of the different <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3445" class="MJXp-math"><span id="MJXp-Span-3446" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-535-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4138" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4139" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4140" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-535" type="math/tex">G</script></span>-conjugates of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3447" class="MJXp-math"><span id="MJXp-Span-3448" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3449" class="MJXp-mi MJXp-italic">ε</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3450" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3451" class="MJXp-mo">(</span><span id="MJXp-Span-3452" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3453" class="MJXp-mo">,</span><span id="MJXp-Span-3454" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3455" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-536-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4141" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4142" class="mrow"><span id="MathJax-Span-4143" class="msubsup"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4144" class="mi">ε</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4145" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4146" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4147" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4148" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4149" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4150" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-536" type="math/tex">\varepsilon_C(K,H)</script></span>. Then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3456" class="MJXp-math"><span id="MJXp-Span-3457" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3458" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3459" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3460" class="MJXp-mo">(</span><span id="MJXp-Span-3461" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3462" class="MJXp-mo">,</span><span id="MJXp-Span-3463" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3464" class="MJXp-mo">,</span><span id="MJXp-Span-3465" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3466" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-537-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4151" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4152" class="mrow"><span id="MathJax-Span-4153" class="msubsup"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4154" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4155" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4156" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4157" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4158" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4159" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4160" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4161" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4162" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-537" type="math/tex">e_C(G,K,H)</script></span> is a <em>primitive central idempotent</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3467" class="MJXp-math"><span id="MJXp-Span-3468" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-3469" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-538-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4163" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4164" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4165" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4166" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-538" type="math/tex">FG</script></span> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBBR">[BR07]</a>. We say that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3470" class="MJXp-math"><span id="MJXp-Span-3471" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3472" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3473" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3474" class="MJXp-mo">(</span><span id="MJXp-Span-3475" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3476" class="MJXp-mo">,</span><span id="MJXp-Span-3477" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3478" class="MJXp-mo">,</span><span id="MJXp-Span-3479" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3480" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-539-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4167" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4168" class="mrow"><span id="MathJax-Span-4169" class="msubsup"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4170" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4171" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4172" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4173" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4174" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4175" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4176" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4177" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4178" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-539" type="math/tex">e_C(G,K,H)</script></span> is the primitive central idempotent realized by the strong Shoda pair <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3481" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3482" class="MJXp-mo">(</span><span id="MJXp-Span-3483" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3484" class="MJXp-mo">,</span><span id="MJXp-Span-3485" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3486" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-540-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4179" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4180" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4181" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4182" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4183" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4184" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4185" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-540" type="math/tex">(K,H)</script></span> of the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3487" class="MJXp-math"><span id="MJXp-Span-3488" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-541-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4186" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4187" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4188" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-541" type="math/tex">G</script></span> and the cyclotomic class <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3489" class="MJXp-math"><span id="MJXp-Span-3490" class="MJXp-mi MJXp-italic">C</span></span></span><span style="" tabindex="0" id="MathJax-Element-542-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4189" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4190" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4191" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-542" type="math/tex">C</script></span>.</p>
-
-<p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3491" class="MJXp-math"><span id="MJXp-Span-3492" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-543-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4192" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4193" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4194" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-543" type="math/tex">G</script></span> is <em>strongly monomial</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84C694978557EFE5"><span class="RefLink">9.16</span></a>) then every primitive central idempotent of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3493" class="MJXp-math"><span id="MJXp-Span-3494" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-3495" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-544-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4195" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4196" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4197" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4198" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-544" type="math/tex">FG</script></span> is realizable by some <em>strong Shoda pair</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3496" class="MJXp-math"><span id="MJXp-Span-3497" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-545-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4199" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4200" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4201" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-545" type="math/tex">G</script></span> and some cyclotomic class <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3498" class="MJXp-math"><span id="MJXp-Span-3499" class="MJXp-mi MJXp-italic">C</span></span></span><span style="" tabindex="0" id="MathJax-Element-546-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4202" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4203" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4204" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-546" type="math/tex">C</script></span> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBBR">[BR07]</a>. As in the zero characteristic case, this explain how to compute the <em>Wedderburn decomposition</em> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3500" class="MJXp-math"><span id="MJXp-Span-3501" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-3502" class="MJXp-mi MJXp-italic">G</span></span></span><span style="" tabindex="0" id="MathJax-Element-547-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4205" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4206" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4207" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4208" class="mi">G</span></span></span></span></span></nobr></span><script id="MathJax-Element-547" type="math/tex">FG</script></span> for a finite semisimple algebra of a strongly monomial group (see <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBBR">[BR07]</a>
- for details). For non strongly monomial groups the algorithm to compute
- the Wedderburn decomposition just uses the Brauer characters.</p>
-
-<p>.</p>
-
-<p><a id="X803562E087325AF6" name="X803562E087325AF6"></a></p>
-
-<h4>9.18 <span class="Heading">Theory for Local Schur Index and Division Algebra Part Calculations</span></h4>
-
-<p>(By Allen Herman, May 2013. Updated October 2014.)</p>
-
-<p>The division algebra parts of simple algebras in the Wedderburn
-Decomposition of the group algebra of a finite group over an abelian
-number field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3503" class="MJXp-math"><span id="MJXp-Span-3504" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-548-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4209" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4210" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4211" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-548" type="math/tex">F</script></span> correspond to elements of the Schur Subgroup <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3505" class="MJXp-math"><span id="MJXp-Span-3506" class="MJXp-mi MJXp-italic">S</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3507" class="MJXp-mo">(</span><span id="MJXp-Span-3508" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3509" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-549-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4212" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4213" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4214" class="mi">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4215" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4216" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4217" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-549" type="math/tex">S(F)</script></span> of the Brauer group of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3510" class="MJXp-math"><span id="MJXp-Span-3511" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-550-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4218" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4219" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4220" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-550" type="math/tex">F</script></span>. Like all classes in the Brauer group of an algebraic number field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3512" class="MJXp-math"><span id="MJXp-Span-3513" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-551-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4221" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4222" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4223" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-551" type="math/tex">F</script></span>, the division algebra part of a representative of a given Brauer class is determined up to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3514" class="MJXp-math"><span id="MJXp-Span-3515" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-552-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4224" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4225" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4226" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-552" type="math/tex">F</script></span>-algebra isomorphism by its list of local Hasse invariants at all primes (i.e. places) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3516" class="MJXp-math"><span id="MJXp-Span-3517" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-553-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4227" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4228" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4229" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-553" type="math/tex">F</script></span>. The local invariant at a prime <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3518" class="MJXp-math"><span id="MJXp-Span-3519" class="MJXp-mi MJXp-italic">P</span></span></span><span style="" tabindex="0" id="MathJax-Element-554-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4230" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4231" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4232" class="mi">P</span></span></span></span></span></nobr></span><script id="MathJax-Element-554" type="math/tex">P</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3520" class="MJXp-math"><span id="MJXp-Span-3521" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-555-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4233" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4234" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4235" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-555" type="math/tex">F</script></span> is a lowest terms fraction <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3522" class="MJXp-math"><span id="MJXp-Span-3523" class="MJXp-mi MJXp-italic">r</span><span id="MJXp-Span-3524" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3525" class="MJXp-mo">/</span></span><span id="MJXp-Span-3526" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3527" class="MJXp-mi MJXp-italic">m</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3528" class="MJXp-mi MJXp-italic MJXp-script">P</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-556-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4236" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4237" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4238" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span id="MathJax-Span-4239" class="texatom"><span id="MathJax-Span-4240" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4241" class="mo">/</span></span></span><span id="MathJax-Span-4242" class="msubsup"><span style="display: inline-block; position: relative; width: 1.229em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.7em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4243" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4244" class="mi">P</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-556" type="math/tex">r/m_P</script></span> whose denominator is the local Schur index <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3529" class="MJXp-math"><span id="MJXp-Span-3530" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3531" class="MJXp-mi MJXp-italic">m</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3532" class="MJXp-mi MJXp-italic MJXp-script">P</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-557-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4245" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4246" class="mrow"><span id="MathJax-Span-4247" class="msubsup"><span style="display: inline-block; position: relative; width: 1.229em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.7em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4248" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4249" class="mi">P</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-557" type="math/tex">m_P</script></span> of the simple algebra at the prime <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3533" class="MJXp-math"><span id="MJXp-Span-3534" class="MJXp-mi MJXp-italic">q</span></span></span><span style="" tabindex="0" id="MathJax-Element-558-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4250" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4251" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4252" class="mi">q</span></span></span></span></span></nobr></span><script id="MathJax-Element-558" type="math/tex">q</script></span> (see <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBR">[Rei03]</a>). For division algebras whose Brauer class lies in the Schur Subgroup of an abelian number field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3535" class="MJXp-math"><span id="MJXp-Span-3536" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-559-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4253" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4254" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4255" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-559" type="math/tex">F</script></span>, the local indices at any of the primes <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3537" class="MJXp-math"><span id="MJXp-Span-3538" class="MJXp-mi MJXp-italic">P</span></span></span><span style="" tabindex="0" id="MathJax-Element-560-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4256" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4257" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4258" class="mi">P</span></span></span></span></span></nobr></span><script id="MathJax-Element-560" type="math/tex">P</script></span> lying over the same rational prime <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3539" class="MJXp-math"><span id="MJXp-Span-3540" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-561-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4259" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4260" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4261" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-561" type="math/tex">p</script></span> are equal to the same positive integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3541" class="MJXp-math"><span id="MJXp-Span-3542" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3543" class="MJXp-mi MJXp-italic">m</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3544" class="MJXp-mi MJXp-italic MJXp-script">p</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-562-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4262" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4263" class="mrow"><span id="MathJax-Span-4264" class="msubsup"><span style="display: inline-block; position: relative; width: 1.153em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.7em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4265" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4266" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-562" type="math/tex">m_p</script></span>, and the numerator of the local invariants among these primes are uniformly distributed among the integers <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3545" class="MJXp-math"><span id="MJXp-Span-3546" class="MJXp-mi MJXp-italic">r</span></span></span><span style="" tabindex="0" id="MathJax-Element-563-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4267" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4268" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4269" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-563" type="math/tex">r</script></span> coprime to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3547" class="MJXp-math"><span id="MJXp-Span-3548" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3549" class="MJXp-mi MJXp-italic">m</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3550" class="MJXp-mi MJXp-italic MJXp-script">p</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-564-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4270" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4271" class="mrow"><span id="MathJax-Span-4272" class="msubsup"><span style="display: inline-block; position: relative; width: 1.153em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.7em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4273" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4274" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-564" type="math/tex">m_p</script></span> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBBS">[BS72]</a>.</p>
-
-<p>The local Schur index functions in wedderga produce a list of the
-nontrivial local indices of the division algebra part of the simple
-algebra at all rational primes. The Schur index of the simple algebra
-over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3551" class="MJXp-math"><span id="MJXp-Span-3552" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-565-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4275" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4276" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4277" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-565" type="math/tex">F</script></span> is the least common multiple <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3553" class="MJXp-math"><span id="MJXp-Span-3554" class="MJXp-mi MJXp-italic">m</span></span></span><span style="" tabindex="0" id="MathJax-Element-566-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4278" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4279" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4280" class="mi">m</span></span></span></span></span></nobr></span><script id="MathJax-Element-566" type="math/tex">m</script></span> of these local indices, and the dimension of the division algebra part of the simple algebra over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3555" class="MJXp-math"><span id="MJXp-Span-3556" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-567-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4281" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4282" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4283" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-567" type="math/tex">F</script></span> is <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3557" class="MJXp-math"><span id="MJXp-Span-3558" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3559" class="MJXp-mi MJXp-italic">m</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3560" class="MJXp-mn MJXp-script">2</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-568-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4284" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4285" class="mrow"><span id="MathJax-Span-4286" class="msubsup"><span style="display: inline-block; position: relative; width: 1.151em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.7em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4287" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-4288" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-568" type="math/tex">m^2</script></span>.
- While not sufficient to identify these division algebras up to ring
-isomorphism in general, this list of local indices does identify the
-division algebra up to ring isomorphism whenever there is no pair of
-local indices at odd primes that are greater than 2. (This is at least
-the case for groups of order less than 3^2*7*13.) So it gives the
-information desired in most basic situations, and allows one to
-distinguish almost all pairs of simple components of group algebras.</p>
-
-<p>Wedderga's functions compute local indices for generalized quaternion
- algebras defined over the rationals and cyclotomic algebras defined
-over any abelian number field. Special shortcut functions are available
-for cyclic cyclotomic algebras. There are also versions of the functions
- that compute the local and global Schur index of a character of a
-finite group over a given abelian number field. The steps in the general
- character- theoretic method involve 1) a Brauer-Witt reduction to a
-cyclic-by-abelian group, 2) use of the Frobenius-Schur indicator to
-compute the local index at infinity, 3) computing the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3561" class="MJXp-math"><span id="MJXp-Span-3562" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-569-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4289" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4290" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4291" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-569" type="math/tex">p</script></span>-local index for an ordinary irreducible character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3563" class="MJXp-math"><span id="MJXp-Span-3564" class="MJXp-mi MJXp-italic">χ</span></span></span><span style="" tabindex="0" id="MathJax-Element-570-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4292" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4293" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4294" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-570" type="math/tex">\chi</script></span> of a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3565" class="MJXp-math"><span id="MJXp-Span-3566" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-571-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4295" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4296" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4297" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-571" type="math/tex">p</script></span>-solvable group using the values of an irreducible Brauer character in the same <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3567" class="MJXp-math"><span id="MJXp-Span-3568" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-572-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4298" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4299" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4300" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-572" type="math/tex">p</script></span>-block in cases where the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3569" class="MJXp-math"><span id="MJXp-Span-3570" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-573-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4301" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4302" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4303" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-573" type="math/tex">p</script></span>-defect group of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3571" class="MJXp-math"><span id="MJXp-Span-3572" class="MJXp-mi MJXp-italic">χ</span></span></span><span style="" tabindex="0" id="MathJax-Element-574-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4304" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4305" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4306" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-574" type="math/tex">\chi</script></span> is cyclic, and 4) use of Riese and Schmid's characterization of dyadic Schur groups (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBSch">[Sch94]</a> and <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBRSch">[RS96]</a>)
- to handle the exceptional cases where step 3) is not available. Our
-approach to rational quaternion algebras is the standard one given, for
-example, in <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBPi">[Pie82]</a>.
- The Legendre symbol operation in GAP is used to determine the local
-index at odd primes. The local index of the generalized quaternion
-algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3573" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3574" class="MJXp-mo">(</span><span id="MJXp-Span-3575" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3576" class="MJXp-mo">,</span><span id="MJXp-Span-3577" class="MJXp-mi MJXp-italic">b</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3578" class="MJXp-mo">)</span></span></span><span style="" tabindex="0" id="MathJax-Element-575-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4307" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4308" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4309" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4310" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4311" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4312" class="mi">b</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4313" class="mo">)</span></span></span></span></span></nobr></span><script id="MathJax-Element-575" type="math/tex">(a,b)</script></span> over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3579" class="MJXp-math"><span id="MJXp-Span-3580" class="MJXp-mi MJXp-italic">Q</span></span></span><span style="" tabindex="0" id="MathJax-Element-576-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4314" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4315" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4316" class="mi">Q</span></span></span></span></span></nobr></span><script id="MathJax-Element-576" type="math/tex">Q</script></span> at the infinite prime will be <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3581" class="MJXp-math"><span id="MJXp-Span-3582" class="MJXp-mn">2</span></span></span><span style="" tabindex="0" id="MathJax-Element-577-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4317" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4318" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4319" class="mn">2</span></span></span></span></span></nobr></span><script id="MathJax-Element-577" type="math/tex">2</script></span> if both <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3583" class="MJXp-math"><span id="MJXp-Span-3584" class="MJXp-mi MJXp-italic">a</span></span></span><span style="" tabindex="0" id="MathJax-Element-578-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4320" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4321" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4322" class="mi">a</span></span></span></span></span></nobr></span><script id="MathJax-Element-578" type="math/tex">a</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3585" class="MJXp-math"><span id="MJXp-Span-3586" class="MJXp-mi MJXp-italic">b</span></span></span><span style="" tabindex="0" id="MathJax-Element-579-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4323" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4324" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4325" class="mi">b</span></span></span></span></span></nobr></span><script id="MathJax-Element-579" type="math/tex">b</script></span> are negative, and otherwise <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3587" class="MJXp-math"><span id="MJXp-Span-3588" class="MJXp-mn">1</span></span></span><span style="" tabindex="0" id="MathJax-Element-580-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4326" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4327" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4328" class="mn">1</span></span></span></span></span></nobr></span><script id="MathJax-Element-580" type="math/tex">1</script></span>. We avoid the complicated case of quadratic reciprocity when working over Q by using the Hasse-Brauer-Albert-Noether Theorem (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBR">[Rei03]</a>, pg. 276): since we know the other primes of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3589" class="MJXp-math"><span id="MJXp-Span-3590" class="MJXp-mi MJXp-italic">Q</span></span></span><span style="" tabindex="0" id="MathJax-Element-581-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4329" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4330" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4331" class="mi">Q</span></span></span></span></span></nobr></span><script id="MathJax-Element-581" type="math/tex">Q</script></span> where the local index is <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3591" class="MJXp-math"><span id="MJXp-Span-3592" class="MJXp-mn">2</span></span></span><span style="" tabindex="0" id="MathJax-Element-582-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4332" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4333" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4334" class="mn">2</span></span></span></span></span></nobr></span><script id="MathJax-Element-582" type="math/tex">2</script></span>, it determines the local index at the prime <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3593" class="MJXp-math"><span id="MJXp-Span-3594" class="MJXp-mn">2</span></span></span><span style="" tabindex="0" id="MathJax-Element-583-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4335" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4336" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4337" class="mn">2</span></span></span></span></span></nobr></span><script id="MathJax-Element-583" type="math/tex">2</script></span>. For generalized quaternion algebras over number fields <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3595" class="MJXp-math"><span id="MJXp-Span-3596" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-584-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4338" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4339" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4340" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-584" type="math/tex">F</script></span> other than <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3597" class="MJXp-math"><span id="MJXp-Span-3598" class="MJXp-mi MJXp-italic">Q</span></span></span><span style="" tabindex="0" id="MathJax-Element-585-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4341" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4342" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4343" class="mi">Q</span></span></span></span></span></nobr></span><script id="MathJax-Element-585" type="math/tex">Q</script></span>,
- we have to convert to cyclic or cyclic cyclotomic algebras and use the
-other local index functions, or appeal to a number theory system outside
- of GAP that can solve norm equations.</p>
-
-<p>There are three shortcut functions used to compute local indices of
-cyclic cyclotomic algebras, which wedderga's -Info functions produce in
-the form <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3599" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3600" class="MJXp-mo">[</span><span id="MJXp-Span-3601" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3602" class="MJXp-mo">,</span><span id="MJXp-Span-3603" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3604" class="MJXp-mo">,</span><span id="MJXp-Span-3605" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3606" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3607" class="MJXp-mo">[</span><span id="MJXp-Span-3608" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3609" class="MJXp-mo">,</span><span id="MJXp-Span-3610" class="MJXp-mi MJXp-italic">b</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3611" class="MJXp-mo">,</span><span id="MJXp-Span-3612" class="MJXp-mi MJXp-italic">c</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3613" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3614" class="MJXp-mo">]</span></span></span><span style="" tabindex="0" id="MathJax-Element-586-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4344" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4345" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4346" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4347" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4348" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4349" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4350" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4351" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4352" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-4353" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4354" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4355" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4356" class="mi">b</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4357" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4358" class="mi">c</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4359" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4360" class="mo">]</span></span></span></span></span></nobr></span><script id="MathJax-Element-586" type="math/tex">[r,F,n,[a,b,c]]</script></span>.
- The local index at infinity is calculated by determining if the real
-completion of the corresponding algebra will produce a real quaternion
-algebra. In order to do this, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3615" class="MJXp-math"><span id="MJXp-Span-3616" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-587-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4361" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4362" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4363" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-587" type="math/tex">F</script></span> must be a real subfield, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3617" class="MJXp-math"><span id="MJXp-Span-3618" class="MJXp-mi MJXp-italic">n</span></span></span><span style="" tabindex="0" id="MathJax-Element-588-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4364" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4365" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4366" class="mi">n</span></span></span></span></span></nobr></span><script id="MathJax-Element-588" type="math/tex">n</script></span> must be strictly greater than <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3619" class="MJXp-math"><span id="MJXp-Span-3620" class="MJXp-mn">2</span></span></span><span style="" tabindex="0" id="MathJax-Element-589-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4367" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4368" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4369" class="mn">2</span></span></span></span></span></nobr></span><script id="MathJax-Element-589" type="math/tex">2</script></span>, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3621" class="MJXp-math"><span id="MJXp-Span-3622" class="MJXp-mi MJXp-italic">E</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3623" class="MJXp-mo">(</span><span id="MJXp-Span-3624" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-3625" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-3626" class="MJXp-mo">)</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3627" class="MJXp-mi MJXp-italic MJXp-script">c</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-590-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4370" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4371" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4372" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4373" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4374" class="mi">n</span><span id="MathJax-Span-4375" class="msubsup"><span style="display: inline-block; position: relative; width: 0.722em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-4376" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4377" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-590" type="math/tex">E(n)^c</script></span> (which has to be a root of unity in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3628" class="MJXp-math"><span id="MJXp-Span-3629" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-591-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4378" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4379" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4380" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-591" type="math/tex">F</script></span>) must be <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3630" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0.111em;" id="MJXp-Span-3631" class="MJXp-mo">−</span><span id="MJXp-Span-3632" class="MJXp-mn">1</span></span></span><span style="" tabindex="0" id="MathJax-Element-592-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4381" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4382" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4383" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4384" class="mn">1</span></span></span></span></span></nobr></span><script id="MathJax-Element-592" type="math/tex">-1</script></span>.
- These facts can be checked directly, so this is faster than calculating
- the character table of the group and checking the value of a
-Frobenius-Schur indicator. The shortcut to calculate the local index of a
- cyclic cyclotomic algebra at an odd prime makes direct use of the
-following lemma of Janusz: If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3633" class="MJXp-math"><span id="MJXp-Span-3634" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3635" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3636" class="MJXp-mi MJXp-italic MJXp-script">p</span></span><span id="MJXp-Span-3637" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3638" class="MJXp-mo">/</span></span><span id="MJXp-Span-3639" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3640" class="MJXp-mi MJXp-italic">F</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3641" class="MJXp-mi MJXp-italic MJXp-script">p</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-593-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4385" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4386" class="mrow"><span id="MathJax-Span-4387" class="msubsup"><span style="display: inline-block; position: relative; width: 1.042em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.63em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4388" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4389" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-4390" class="texatom"><span id="MathJax-Span-4391" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4392" class="mo">/</span></span></span><span id="MathJax-Span-4393" class="msubsup"><span style="display: inline-block; position: relative; width: 1.042em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.65em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4394" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4395" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-593" type="math/tex">E_p/F_p</script></span> is a Galois extension of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3642" class="MJXp-math"><span id="MJXp-Span-3643" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-594-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4396" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4397" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4398" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-594" type="math/tex">p</script></span>-local fields with ramification index <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3644" class="MJXp-math"><span id="MJXp-Span-3645" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-595-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4399" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4400" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4401" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-595" type="math/tex">e</script></span>, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3646" class="MJXp-math"><span id="MJXp-Span-3647" class="MJXp-mi MJXp-italic">z</span></span></span><span style="" tabindex="0" id="MathJax-Element-596-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4402" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4403" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4404" class="mi">z</span></span></span></span></span></nobr></span><script id="MathJax-Element-596" type="math/tex">z</script></span> is a root of unity with order prime to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3648" class="MJXp-math"><span id="MJXp-Span-3649" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-597-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4405" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4406" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4407" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-597" type="math/tex">p</script></span>, then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3650" class="MJXp-math"><span id="MJXp-Span-3651" class="MJXp-mi MJXp-italic">z</span></span></span><span style="" tabindex="0" id="MathJax-Element-598-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4408" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4409" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4410" class="mi">z</span></span></span></span></span></nobr></span><script id="MathJax-Element-598" type="math/tex">z</script></span> is a norm in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3652" class="MJXp-math"><span id="MJXp-Span-3653" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3654" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3655" class="MJXp-mi MJXp-italic MJXp-script">p</span></span><span id="MJXp-Span-3656" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3657" class="MJXp-mo">/</span></span><span id="MJXp-Span-3658" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3659" class="MJXp-mi MJXp-italic">F</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3660" class="MJXp-mi MJXp-italic MJXp-script">p</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-599-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4411" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4412" class="mrow"><span id="MathJax-Span-4413" class="msubsup"><span style="display: inline-block; position: relative; width: 1.042em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.63em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4414" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4415" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-4416" class="texatom"><span id="MathJax-Span-4417" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4418" class="mo">/</span></span></span><span id="MathJax-Span-4419" class="msubsup"><span style="display: inline-block; position: relative; width: 1.042em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.65em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4420" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4421" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-599" type="math/tex">E_p/F_p</script></span> if and only if it is the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3661" class="MJXp-math"><span id="MJXp-Span-3662" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-600-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4422" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4423" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4424" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-600" type="math/tex">e</script></span>-th power of a root of unity in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3663" class="MJXp-math"><span id="MJXp-Span-3664" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-601-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4425" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4426" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4427" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-601" type="math/tex">F</script></span>. (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBJ">[Jan75]</a>, pg. 535). It follows that in order to calculate the local index at <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3665" class="MJXp-math"><span id="MJXp-Span-3666" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-602-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4428" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4429" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4430" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-602" type="math/tex">p</script></span> of a cyclic cyclotomic algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3667" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3668" class="MJXp-mo">[</span><span id="MJXp-Span-3669" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3670" class="MJXp-mo">,</span><span id="MJXp-Span-3671" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3672" class="MJXp-mo">,</span><span id="MJXp-Span-3673" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3674" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3675" class="MJXp-mo">[</span><span id="MJXp-Span-3676" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3677" class="MJXp-mo">,</span><span id="MJXp-Span-3678" class="MJXp-mi MJXp-italic">b</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3679" class="MJXp-mo">,</span><span id="MJXp-Span-3680" class="MJXp-mi MJXp-italic">c</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3681" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3682" class="MJXp-mo">]</span></span></span><span style="" tabindex="0" id="MathJax-Element-603-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4431" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4432" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4433" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4434" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4435" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4436" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4437" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4438" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4439" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-4440" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4441" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4442" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4443" class="mi">b</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4444" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4445" class="mi">c</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4446" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4447" class="mo">]</span></span></span></span></span></nobr></span><script id="MathJax-Element-603" type="math/tex">[r,F,n,[a,b,c]]</script></span>, we first determine the splitting degree, residue degree, and ramification index <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3683" class="MJXp-math"><span id="MJXp-Span-3684" class="MJXp-mi MJXp-italic">e</span></span></span><span style="" tabindex="0" id="MathJax-Element-604-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4448" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4449" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4450" class="mi">e</span></span></span></span></span></nobr></span><script id="MathJax-Element-604" type="math/tex">e</script></span> of the extension <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3685" class="MJXp-math"><span id="MJXp-Span-3686" class="MJXp-mi MJXp-italic">F</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3687" class="MJXp-mo">(</span><span id="MJXp-Span-3688" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3689" class="MJXp-mi MJXp-italic">ζ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3690" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3691" class="MJXp-mo">)</span><span id="MJXp-Span-3692" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3693" class="MJXp-mo">/</span></span><span id="MJXp-Span-3694" class="MJXp-mi MJXp-italic">F</span></span></span><span style="" tabindex="0" id="MathJax-Element-605-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4451" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4452" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4453" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4454" class="mo">(</span><span id="MathJax-Span-4455" class="msubsup"><span style="display: inline-block; position: relative; width: 0.883em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.32em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4456" class="mi">ζ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.021em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.454em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4457" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4458" class="mo">)</span><span id="MathJax-Span-4459" class="texatom"><span id="MathJax-Span-4460" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-4461" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4462" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-605" type="math/tex">F(\zeta_n)/F</script></span> at <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3695" class="MJXp-math"><span id="MJXp-Span-3696" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-606-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4463" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4464" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4465" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-606" type="math/tex">p</script></span>. Comparing the behaviour of the Galois automorphism <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3697" class="MJXp-math"><span id="MJXp-Span-3698" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3699" class="MJXp-mi MJXp-italic">σ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3700" class="MJXp-mi MJXp-italic MJXp-script">b</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-607-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4466" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4467" class="mrow"><span id="MathJax-Span-4468" class="msubsup"><span style="display: inline-block; position: relative; width: 0.927em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.53em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4469" class="mi">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.033em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.498em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4470" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-607" type="math/tex">\sigma_b</script></span> to the behaviour of the Frobenius automorphism at <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3701" class="MJXp-math"><span id="MJXp-Span-3702" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-608-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4471" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4472" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4473" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-608" type="math/tex">p</script></span> allows us to determine the order of the largest root of unity <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3703" class="MJXp-math"><span id="MJXp-Span-3704" class="MJXp-mi MJXp-italic">z</span></span></span><span style="" tabindex="0" id="MathJax-Element-609-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4474" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4475" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4476" class="mi">z</span></span></span></span></span></nobr></span><script id="MathJax-Element-609" type="math/tex">z</script></span> with order coprime to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3705" class="MJXp-math"><span id="MJXp-Span-3706" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-610-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4477" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4478" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4479" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-610" type="math/tex">p</script></span> in the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3707" class="MJXp-math"><span id="MJXp-Span-3708" class="MJXp-mi MJXp-italic">p</span></span></span><span style="" tabindex="0" id="MathJax-Element-611-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4480" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4481" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4482" class="mi">p</span></span></span></span></span></nobr></span><script id="MathJax-Element-611" type="math/tex">p</script></span>-completion <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3709" class="MJXp-math"><span id="MJXp-Span-3710" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3711" class="MJXp-mi MJXp-italic">F</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3712" class="MJXp-mi MJXp-italic MJXp-script">p</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-612-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4483" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4484" class="mrow"><span id="MathJax-Span-4485" class="msubsup"><span style="display: inline-block; position: relative; width: 1.042em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.65em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4486" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4487" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-612" type="math/tex">F_p</script></span>. The local index <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3713" class="MJXp-math"><span id="MJXp-Span-3714" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3715" class="MJXp-mi MJXp-italic">m</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3716" class="MJXp-mi MJXp-italic MJXp-script">p</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-613-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4488" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4489" class="mrow"><span id="MathJax-Span-4490" class="msubsup"><span style="display: inline-block; position: relative; width: 1.153em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.7em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4491" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4492" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-613" type="math/tex">m_p</script></span> is then the least power of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3717" class="MJXp-math"><span id="MJXp-Span-3718" class="MJXp-mi MJXp-italic">E</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3719" class="MJXp-mo">(</span><span id="MJXp-Span-3720" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-3721" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-3722" class="MJXp-mo">)</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3723" class="MJXp-mi MJXp-italic MJXp-script">c</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-614-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4493" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4494" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4495" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4496" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4497" class="mi">n</span><span id="MathJax-Span-4498" class="msubsup"><span style="display: inline-block; position: relative; width: 0.722em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-4499" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4500" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-614" type="math/tex">E(n)^c</script></span> that lies in the group generated by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3724" class="MJXp-math"><span id="MJXp-Span-3725" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3726" class="MJXp-mi MJXp-italic">z</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3727" class="MJXp-mi MJXp-italic MJXp-script">e</span></span></span></span><span style="" tabindex="0" id="MathJax-Element-615-Frame" class="MathJax MathJax_Processed"><nobr><span role="math" id="MathJax-Span-4501" class="math"><span style="display: inline-block; position: relative; width: 0em; height: 0px; font-size: 130%;"><span style="position: absolute;"><span id="MathJax-Span-4502" class="mrow"><span id="MathJax-Span-4503" class="msubsup"><span style="display: inline-block; position: relative; width: 0.778em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4504" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4505" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span></nobr></span><script id="MathJax-Element-615" type="math/tex">z^e</script></span>.</p>
-
-<p>Calculation of the local index at the prime <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3728" class="MJXp-math"><span id="MJXp-Span-3729" class="MJXp-mn">2</span></span></span><span tabindex="0" id="MathJax-Element-616-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-616" type="math/tex">2</script></span> makes use of the following consequence of (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBJ">[Jan75]</a>, Theorem 5): A cyclic cyclotomic algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3730" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3731" class="MJXp-mo">[</span><span id="MJXp-Span-3732" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3733" class="MJXp-mo">,</span><span id="MJXp-Span-3734" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3735" class="MJXp-mi MJXp-italic">F</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3736" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3737" class="MJXp-mo">,</span><span id="MJXp-Span-3738" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3739" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3740" class="MJXp-mo">[</span><span id="MJXp-Span-3741" class="MJXp-mi MJXp-italic">a</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3742" class="MJXp-mo">,</span><span id="MJXp-Span-3743" class="MJXp-mi MJXp-italic">b</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3744" class="MJXp-mo">,</span><span id="MJXp-Span-3745" class="MJXp-mi MJXp-italic">c</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3746" class="MJXp-mo">]</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3747" class="MJXp-mo">]</span></span></span><span tabindex="0" id="MathJax-Element-617-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-617" type="math/tex">[r,F_2,n,[a,b,c]]</script></span> over a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3748" class="MJXp-math"><span id="MJXp-Span-3749" class="MJXp-mn">2</span></span></span><span tabindex="0" id="MathJax-Element-618-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-618" type="math/tex">2</script></span>-local field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3750" class="MJXp-math"><span id="MJXp-Span-3751" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3752" class="MJXp-mi MJXp-italic">F</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3753" class="MJXp-mn MJXp-script">2</span></span></span></span><span tabindex="0" id="MathJax-Element-619-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-619" type="math/tex">F_2</script></span> that is a subfield of a cyclotomic extension of the rational <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3754" class="MJXp-math"><span id="MJXp-Span-3755" class="MJXp-mn">2</span></span></span><span tabindex="0" id="MathJax-Element-620-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-620" type="math/tex">2</script></span>-local field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3756" class="MJXp-math"><span id="MJXp-Span-3757" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3758" class="MJXp-mi MJXp-italic">Q</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3759" class="MJXp-mn MJXp-script">2</span></span></span></span><span tabindex="0" id="MathJax-Element-621-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-621" type="math/tex">Q_2</script></span> has Schur index at most <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3760" class="MJXp-math"><span id="MJXp-Span-3761" class="MJXp-mn">2</span></span></span><span tabindex="0" id="MathJax-Element-622-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-622" type="math/tex">2</script></span>. It has Schur index <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3762" class="MJXp-math"><span id="MJXp-Span-3763" class="MJXp-mn">2</span></span></span><span tabindex="0" id="MathJax-Element-623-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-623" type="math/tex">2</script></span> if and only if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3764" class="MJXp-math"><span id="MJXp-Span-3765" class="MJXp-mn">4</span></span></span><span tabindex="0" id="MathJax-Element-624-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-624" type="math/tex">4</script></span> divides <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3766" class="MJXp-math"><span id="MJXp-Span-3767" class="MJXp-mi MJXp-italic">n</span></span></span><span tabindex="0" id="MathJax-Element-625-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-625" type="math/tex">n</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3768" class="MJXp-math"><span id="MJXp-Span-3769" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3770" class="MJXp-mi MJXp-italic">F</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3771" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3772" class="MJXp-mo">(</span><span id="MJXp-Span-3773" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3774" class="MJXp-mi MJXp-italic">ζ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3775" class="MJXp-mn MJXp-script">4</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3776" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-626-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-626" type="math/tex">F_2(\zeta_4)</script></span> is totally ramified of degree 2, the Galois automorphism <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3777" class="MJXp-math"><span id="MJXp-Span-3778" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3779" class="MJXp-mi MJXp-italic">σ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3780" class="MJXp-mi MJXp-italic MJXp-script">b</span></span></span></span><span tabindex="0" id="MathJax-Element-627-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-627" type="math/tex">\sigma_b</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3781" class="MJXp-math"><span id="MJXp-Span-3782" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3783" class="MJXp-mi MJXp-italic">F</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3784" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3785" class="MJXp-mo">(</span><span id="MJXp-Span-3786" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3787" class="MJXp-mi MJXp-italic">ζ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3788" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3789" class="MJXp-mo">)</span><span id="MJXp-Span-3790" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-3791" class="MJXp-mo">/</span></span><span id="MJXp-Span-3792" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3793" class="MJXp-mi MJXp-italic">F</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3794" class="MJXp-mn MJXp-script">2</span></span></span></span><span tabindex="0" id="MathJax-Element-628-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-628" type="math/tex">F_2(\zeta_n)/F_2</script></span> inverts all <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3795" class="MJXp-math"><span id="MJXp-Span-3796" class="MJXp-mn">2</span></span></span><span tabindex="0" id="MathJax-Element-629-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-629" type="math/tex">2</script></span>-power roots of unity in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3797" class="MJXp-math"><span id="MJXp-Span-3798" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3799" class="MJXp-mi MJXp-italic">F</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3800" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3801" class="MJXp-mo">(</span><span id="MJXp-Span-3802" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3803" class="MJXp-mi MJXp-italic">ζ</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3804" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3805" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-630-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-630" type="math/tex">F_2(\zeta_n)</script></span>, the order of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3806" class="MJXp-math"><span id="MJXp-Span-3807" class="MJXp-mi MJXp-italic">E</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3808" class="MJXp-mo">(</span><span id="MJXp-Span-3809" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-3810" class="MJXp-msubsup"><span style="margin-left: 0em; margin-right: 0.05em;" id="MJXp-Span-3811" class="MJXp-mo">)</span><span style="vertical-align: 0.5em;" id="MJXp-Span-3812" class="MJXp-mi MJXp-italic MJXp-script">c</span></span></span></span><span tabindex="0" id="MathJax-Element-631-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-631" type="math/tex">E(n)^c</script></span> is 2 times an odd number, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3813" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3814" class="MJXp-mo">(</span><span id="MJXp-Span-3815" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3816" class="MJXp-mi MJXp-italic">F</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3817" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-3818" class="MJXp-mo">:</span><span id="MJXp-Span-3819" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3820" class="MJXp-mi MJXp-italic">Q</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3821" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3822" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-632-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-632" type="math/tex">(F_2:Q_2)</script></span> is odd. The same approach to cyclotomic reciprocity makes it possible to check all of these conditions in the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3823" class="MJXp-math"><span id="MJXp-Span-3824" class="MJXp-mn">2</span></span></span><span tabindex="0" id="MathJax-Element-633-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-633" type="math/tex">2</script></span>-local situation.</p>
-
-<p>The wedderga function that computes the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3825" class="MJXp-math"><span id="MJXp-Span-3826" class="MJXp-mi MJXp-italic">p</span></span></span><span tabindex="0" id="MathJax-Element-634-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-634" type="math/tex">p</script></span>-local index of an ordinary irreducible character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3827" class="MJXp-math"><span id="MJXp-Span-3828" class="MJXp-mi MJXp-italic">χ</span></span></span><span tabindex="0" id="MathJax-Element-635-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-635" type="math/tex">\chi</script></span> of a finite non-nilpotent cyclic-by- abelian group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3829" class="MJXp-math"><span id="MJXp-Span-3830" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-636-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-636" type="math/tex">G</script></span> is based directly on a theorem of Benard <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBB">[Ben76]</a> that applies whenever the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3831" class="MJXp-math"><span id="MJXp-Span-3832" class="MJXp-mi MJXp-italic">p</span></span></span><span tabindex="0" id="MathJax-Element-637-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-637" type="math/tex">p</script></span>-defect group of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3833" class="MJXp-math"><span id="MJXp-Span-3834" class="MJXp-mi MJXp-italic">χ</span></span></span><span tabindex="0" id="MathJax-Element-638-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-638" type="math/tex">\chi</script></span> is cyclic. We have to restrict our application of it to groups whose orders are small because the <strong class="pkg">GAP</strong>
- records for irreducible Brauer characters are only available in these
-cases. In order to use this approach effectively, we developed a
-function that computes the defect group of the block containing a given
-ordinary irreducible character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3835" class="MJXp-math"><span id="MJXp-Span-3836" class="MJXp-mi MJXp-italic">χ</span></span></span><span tabindex="0" id="MathJax-Element-639-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-639" type="math/tex">\chi</script></span>. This function makes use of the Min half of Brauer's Min-Max theorem (see Theorem 4.4 of <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBN">[Nav98]</a>),
- and thus is able to find the defect group directly from the ordinary
-character table. It is thus available for nonsolvable groups, even in
-cases where <strong class="pkg">GAP</strong>'s Brauer character records
-are not available. We are indebted to Michael Geline and Friederich
-Ladisch for discussions concerning the calculation of defect groups in <strong class="pkg">GAP</strong>. The current algorithm we use is based on an approach suggested by Ladisch.</p>
-
-<p><a id="X7B18AF347AE68020" name="X7B18AF347AE68020"></a></p>
-
-<h4>9.19 <span class="Heading"> Obtaining Algebras with structure constants as terms of the Wedderburn decomposition </span></h4>
-
-<p>Some users may find it desirable to have an alternative description
-for the components of the Wedderburn decomposition of a group ring as
-algebras with structure constants, because the operations for algebras
-in <strong class="pkg">GAP</strong> are designed for algebras with structure constants. We have provided such an algorithm that converts the output of <code class="func">WedderburnDecompositionInfo</code> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap2_mj.html#X8710F98A85F0DD29"><span class="RefLink">2.1-2</span></a>)
- into algebras with structure constants. Matrix rings over fields are
-converted directly. For components that are cyclotomic algebras, it
-calculates their defining group and defining character using those <strong class="pkg">Wedderga</strong> operations, then uses <code class="func">IrreducibleRepresentationsDixon</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8493ED7A86FFCB8A"><span class="RefLink">Reference: IrreducibleRepresentationsDixon</span></a>)
- to obtain matrix generators of an algebra isomorphic to the simple
-component corresponding to the character over a suitable field. An
-algebra with structure constants version of this is finally obtained by
-applying <code class="func">IsomorphismSCAlgebra</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap62_mj.html#X7F8D3DF2863EC50D"><span class="RefLink">Reference: IsomorphismSCAlgebra (w.r.t. a given basis)</span></a>) to this algebra.</p>
-
-<p><a id="X8472ACCF802EC188" name="X8472ACCF802EC188"></a></p>
-
-<h4>9.20 <span class="Heading">A complete set of orthogonal primitive idempotents</span></h4>
-
-<p>When <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3837" class="MJXp-math"><span id="MJXp-Span-3838" class="MJXp-mi MJXp-italic">R</span></span></span><span tabindex="0" id="MathJax-Element-640-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-640" type="math/tex">R</script></span> is a semisimple ring, then every left ideal <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3839" class="MJXp-math"><span id="MJXp-Span-3840" class="MJXp-mi MJXp-italic">L</span></span></span><span tabindex="0" id="MathJax-Element-641-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-641" type="math/tex">L</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3841" class="MJXp-math"><span id="MJXp-Span-3842" class="MJXp-mi MJXp-italic">R</span></span></span><span tabindex="0" id="MathJax-Element-642-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-642" type="math/tex">R</script></span> is of the form <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3843" class="MJXp-math"><span id="MJXp-Span-3844" class="MJXp-mi MJXp-italic">L</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3845" class="MJXp-mo">=</span><span id="MJXp-Span-3846" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-3847" class="MJXp-mi MJXp-italic">e</span></span></span><span tabindex="0" id="MathJax-Element-643-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-643" type="math/tex">L=Re</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3848" class="MJXp-math"><span id="MJXp-Span-3849" class="MJXp-mi MJXp-italic">e</span></span></span><span tabindex="0" id="MathJax-Element-644-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-644" type="math/tex">e</script></span> is an idempotent of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3850" class="MJXp-math"><span id="MJXp-Span-3851" class="MJXp-mi MJXp-italic">R</span></span></span><span tabindex="0" id="MathJax-Element-645-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-645" type="math/tex">R</script></span>.
- Therefore, we can use the idempotents to characterize the
-decompositions of semisimple rings as a direct sum of minimal left
-ideals. In particular, let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3852" class="MJXp-math"><span id="MJXp-Span-3853" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3854" class="MJXp-mo">=</span><span id="MJXp-Span-3855" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-3856" class="MJXp-mo">⊕</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-3861" class="MJXp-mi MJXp-italic">t</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-3857" class="MJXp-mrow"><span id="MJXp-Span-3858" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-3859" class="MJXp-mo">=</span><span id="MJXp-Span-3860" class="MJXp-mn">1</span></span></span></span></span></span></span><span id="MJXp-Span-3862" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3863" class="MJXp-mi MJXp-italic">L</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3864" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span tabindex="0" id="MathJax-Element-646-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-646" type="math/tex">R=\oplus_{i=1}^t L_i</script></span> be a decomposition of a semisimple ring as a direct sum of minimal left ideals. Then, there exists a family <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3865" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3866" class="MJXp-mo">{</span><span id="MJXp-Span-3867" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3868" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3869" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3870" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3871" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3872" class="MJXp-mo">,</span><span id="MJXp-Span-3873" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3874" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3875" class="MJXp-mi MJXp-italic MJXp-script">t</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3876" class="MJXp-mo">}</span></span></span><span tabindex="0" id="MathJax-Element-647-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-647" type="math/tex">\{e_1,\dots,e_t\}</script></span> of elements of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3877" class="MJXp-math"><span id="MJXp-Span-3878" class="MJXp-mi MJXp-italic">R</span></span></span><span tabindex="0" id="MathJax-Element-648-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-648" type="math/tex">R</script></span> such that: each <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3879" class="MJXp-math"><span id="MJXp-Span-3880" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3881" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3882" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3883" class="MJXp-mo">≠</span><span id="MJXp-Span-3884" class="MJXp-mn">0</span></span></span><span tabindex="0" id="MathJax-Element-649-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-649" type="math/tex">e_i\neq 0</script></span> is an idempotent element, if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3885" class="MJXp-math"><span id="MJXp-Span-3886" class="MJXp-mi MJXp-italic">i</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3887" class="MJXp-mo">≠</span><span id="MJXp-Span-3888" class="MJXp-mi MJXp-italic">j</span></span></span><span tabindex="0" id="MathJax-Element-650-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-650" type="math/tex">i\neq j</script></span>, then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3889" class="MJXp-math"><span id="MJXp-Span-3890" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3891" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3892" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span id="MJXp-Span-3893" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3894" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3895" class="MJXp-mi MJXp-italic MJXp-script">j</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3896" class="MJXp-mo">=</span><span id="MJXp-Span-3897" class="MJXp-mn">0</span></span></span><span tabindex="0" id="MathJax-Element-651-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-651" type="math/tex">e_ie_j=0</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3898" class="MJXp-math"><span id="MJXp-Span-3899" class="MJXp-mn">1</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3900" class="MJXp-mo">=</span><span id="MJXp-Span-3901" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3902" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3903" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-3904" class="MJXp-mo">+</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3905" class="MJXp-mo">⋯</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-3906" class="MJXp-mo">+</span><span id="MJXp-Span-3907" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3908" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3909" class="MJXp-mi MJXp-italic MJXp-script">t</span></span></span></span><span tabindex="0" id="MathJax-Element-652-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-652" type="math/tex">1=e_1+\cdots+e_t</script></span> and each <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3910" class="MJXp-math"><span id="MJXp-Span-3911" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3912" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3913" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span tabindex="0" id="MathJax-Element-653-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-653" type="math/tex">e_i</script></span> cannot be written as <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3928" class="MJXp-math"><span id="MJXp-Span-3929" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3930" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3931" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3932" class="MJXp-mo">=</span><span id="MJXp-Span-3933" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3934" class="MJXp-mi MJXp-italic">e</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-3936" class="MJXp-mo">′</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-3935" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-3937" class="MJXp-mo">+</span><span id="MJXp-Span-3938" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3939" class="MJXp-mi MJXp-italic">e</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-3941" class="MJXp-mo">″</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-3940" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-654-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-654" type="math/tex">e_i=e_i'+e_i''</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3942" class="MJXp-math"><span id="MJXp-Span-3943" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3944" class="MJXp-mi MJXp-italic">e</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-3946" class="MJXp-mo">′</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-3945" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3947" class="MJXp-mo">,</span><span id="MJXp-Span-3948" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3949" class="MJXp-mi MJXp-italic">e</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-3951" class="MJXp-mo">″</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-3950" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-655-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-655" type="math/tex">e_i',e_i''</script></span> are idempotents such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3952" class="MJXp-math"><span id="MJXp-Span-3953" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3954" class="MJXp-mi MJXp-italic">e</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-3956" class="MJXp-mo">′</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-3955" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3957" class="MJXp-mo">,</span><span id="MJXp-Span-3958" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3959" class="MJXp-mi MJXp-italic">e</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-3961" class="MJXp-mo">″</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-3960" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3962" class="MJXp-mo">≠</span><span id="MJXp-Span-3963" class="MJXp-mn">0</span></span></span><span tabindex="0" id="MathJax-Element-656-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-656" type="math/tex">e_i',e_i''\neq 0</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3964" class="MJXp-math"><span id="MJXp-Span-3965" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3966" class="MJXp-mi MJXp-italic">e</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-3968" class="MJXp-mo">′</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-3967" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span><span id="MJXp-Span-3969" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3970" class="MJXp-mi MJXp-italic">e</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-3972" class="MJXp-mo">″</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-3971" class="MJXp-mi MJXp-italic">i</span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3973" class="MJXp-mo">=</span><span id="MJXp-Span-3974" class="MJXp-mn">0</span></span></span><span tabindex="0" id="MathJax-Element-657-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-657" type="math/tex">e_i'e_i''=0</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3975" class="MJXp-math"><span id="MJXp-Span-3976" class="MJXp-mn">1</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3977" class="MJXp-mo">≤</span><span id="MJXp-Span-3978" class="MJXp-mi MJXp-italic">i</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3979" class="MJXp-mo">≤</span></span></span><span tabindex="0" id="MathJax-Element-658-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-658" type="math/tex">1\leq i\leq </script></span>. Conversely, if there exists a family of idempotents <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3980" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3981" class="MJXp-mo">{</span><span id="MJXp-Span-3982" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3983" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3984" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3985" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3986" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-3987" class="MJXp-mo">,</span><span id="MJXp-Span-3988" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3989" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3990" class="MJXp-mi MJXp-italic MJXp-script">t</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-3991" class="MJXp-mo">}</span></span></span><span tabindex="0" id="MathJax-Element-659-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-659" type="math/tex">\{e_1,\dots,e_t\}</script></span> satisfying the previous conditions, then the left ideals <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-3992" class="MJXp-math"><span id="MJXp-Span-3993" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3994" class="MJXp-mi MJXp-italic">L</span><span style="vertical-align: -0.4em;" id="MJXp-Span-3995" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-3996" class="MJXp-mo">=</span><span id="MJXp-Span-3997" class="MJXp-mi MJXp-italic">R</span><span id="MJXp-Span-3998" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-3999" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4000" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span tabindex="0" id="MathJax-Element-660-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-660" type="math/tex">L_i=Re_i</script></span> are minimal and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4001" class="MJXp-math"><span id="MJXp-Span-4002" class="MJXp-mi MJXp-italic">R</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4003" class="MJXp-mo">=</span><span id="MJXp-Span-4004" class="MJXp-msubsup"><span style="margin-left: 0.267em; margin-right: 0.05em;" id="MJXp-Span-4005" class="MJXp-mo">⊕</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4010" class="MJXp-mi MJXp-italic">t</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-4006" class="MJXp-mrow"><span id="MJXp-Span-4007" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-4008" class="MJXp-mo">=</span><span id="MJXp-Span-4009" class="MJXp-mn">1</span></span></span></span></span></span></span><span id="MJXp-Span-4011" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4012" class="MJXp-mi MJXp-italic">L</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4013" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span tabindex="0" id="MathJax-Element-661-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-661" type="math/tex">R=\oplus_{i=1}^t L_i</script></span>. Such a set of idempotents is called a <em>complete set of orthogonal primitive idempotents</em> of the ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4014" class="MJXp-math"><span id="MJXp-Span-4015" class="MJXp-mi MJXp-italic">R</span></span></span><span tabindex="0" id="MathJax-Element-662-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-662" type="math/tex">R</script></span>. Such a set is not uniquely determined.</p>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4016" class="MJXp-math"><span id="MJXp-Span-4017" class="MJXp-mrow"><span id="MJXp-Span-4018" class="MJXp-mi undefined">F</span></span></span></span><span tabindex="0" id="MathJax-Element-663-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-663" type="math/tex">\mathbb F</script></span> be a finite field and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4019" class="MJXp-math"><span id="MJXp-Span-4020" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-664-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-664" type="math/tex">G</script></span> a finite nilpotent group such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4021" class="MJXp-math"><span id="MJXp-Span-4022" class="MJXp-mrow"><span id="MJXp-Span-4023" class="MJXp-mi undefined">F</span></span><span id="MJXp-Span-4024" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-665-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-665" type="math/tex">\mathbb F G</script></span> is semisimple. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4025" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4026" class="MJXp-mo">(</span><span id="MJXp-Span-4027" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4028" class="MJXp-mo">,</span><span id="MJXp-Span-4029" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4030" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-666-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-666" type="math/tex">(H,K)</script></span> be a strong Shoda pair of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4031" class="MJXp-math"><span id="MJXp-Span-4032" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-667-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-667" type="math/tex">G</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4033" class="MJXp-math"><span id="MJXp-Span-4034" class="MJXp-mi MJXp-italic">C</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4035" class="MJXp-mo">∈</span><span id="MJXp-Span-4036" class="MJXp-mrow"><span id="MJXp-Span-4037" class="MJXp-mi MJXp-cal">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4038" class="MJXp-mo">(</span><span id="MJXp-Span-4039" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-4040" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4041" class="MJXp-mo">/</span></span><span id="MJXp-Span-4042" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4043" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-668-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-668" type="math/tex">C\in\mathcal{C}(H/K)</script></span> and set <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4044" class="MJXp-math"><span id="MJXp-Span-4045" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4046" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4047" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4048" class="MJXp-mo">=</span><span id="MJXp-Span-4049" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4050" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4051" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4052" class="MJXp-mo">(</span><span id="MJXp-Span-4053" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4054" class="MJXp-mo">,</span><span id="MJXp-Span-4055" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4056" class="MJXp-mo">,</span><span id="MJXp-Span-4057" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4058" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-669-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-669" type="math/tex">e_C=e_C(G,H,K)</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4059" class="MJXp-math"><span id="MJXp-Span-4060" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4061" class="MJXp-mi MJXp-italic">ε</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4062" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4063" class="MJXp-mo">=</span><span id="MJXp-Span-4064" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4065" class="MJXp-mi MJXp-italic">ε</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4066" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4067" class="MJXp-mo">(</span><span id="MJXp-Span-4068" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4069" class="MJXp-mo">,</span><span id="MJXp-Span-4070" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4071" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-670-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-670" type="math/tex">\varepsilon_C=\varepsilon_C(H,K)</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4072" class="MJXp-math"><span id="MJXp-Span-4073" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-4074" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4075" class="MJXp-mo">/</span></span><span id="MJXp-Span-4076" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4077" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4078" class="MJXp-mo">⟨</span><span id="MJXp-Span-4079" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4081" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4080" class="MJXp-mi MJXp-italic">a</span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4082" class="MJXp-mo">⟩</span></span></span><span tabindex="0" id="MathJax-Element-671-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-671" type="math/tex">H/K=\langle\overline{a}\rangle</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4083" class="MJXp-math"><span id="MJXp-Span-4084" class="MJXp-mi MJXp-italic">E</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4085" class="MJXp-mo">=</span><span id="MJXp-Span-4086" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4087" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4088" class="MJXp-mi MJXp-italic MJXp-script">G</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4089" class="MJXp-mo">(</span><span id="MJXp-Span-4090" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-4091" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4092" class="MJXp-mo">/</span></span><span id="MJXp-Span-4093" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4094" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-672-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-672" type="math/tex">E=E_G(H/K)</script></span>. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4095" class="MJXp-math"><span id="MJXp-Span-4096" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4097" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4098" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4099" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4100" class="MJXp-mo">/</span></span><span id="MJXp-Span-4101" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-673-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-673" type="math/tex">E_2/K</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4102" class="MJXp-math"><span id="MJXp-Span-4103" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4104" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4105" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4106" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4107" class="MJXp-mo">/</span></span><span id="MJXp-Span-4108" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4109" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4110" class="MJXp-mo">⟨</span><span id="MJXp-Span-4111" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4115" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4112" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4113" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4114" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4116" class="MJXp-mo">⟩</span></span></span><span tabindex="0" id="MathJax-Element-674-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-674" type="math/tex">H_2/K=\langle\overline{a_2}\rangle</script></span> (respectively <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4117" class="MJXp-math"><span id="MJXp-Span-4118" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4119" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4120" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4121" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4122" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4123" class="MJXp-mo MJXp-script">′</span></span></span></span><span id="MJXp-Span-4124" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4125" class="MJXp-mo">/</span></span><span id="MJXp-Span-4126" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-675-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-675" type="math/tex">E_{2'}/K</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4127" class="MJXp-math"><span id="MJXp-Span-4128" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4129" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4130" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4131" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4132" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4133" class="MJXp-mo MJXp-script">′</span></span></span></span><span id="MJXp-Span-4134" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4135" class="MJXp-mo">/</span></span><span id="MJXp-Span-4136" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4137" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4138" class="MJXp-mo">⟨</span><span id="MJXp-Span-4139" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4146" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4140" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4141" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4142" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4143" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4144" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4145" class="MJXp-mo MJXp-script">′</span></span></span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4147" class="MJXp-mo">⟩</span></span></span><span tabindex="0" id="MathJax-Element-676-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-676" type="math/tex">H_{2'}/K=\langle\overline{a_{2'}}\rangle</script></span>) denote the 2-parts (respectively 2'-parts) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4148" class="MJXp-math"><span id="MJXp-Span-4149" class="MJXp-mi MJXp-italic">E</span><span id="MJXp-Span-4150" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4151" class="MJXp-mo">/</span></span><span id="MJXp-Span-4152" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-677-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-677" type="math/tex">E/K</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4153" class="MJXp-math"><span id="MJXp-Span-4154" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-4155" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4156" class="MJXp-mo">/</span></span><span id="MJXp-Span-4157" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-678-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-678" type="math/tex">H/K</script></span> respectively. Then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4158" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4159" class="MJXp-mo">⟨</span><span id="MJXp-Span-4160" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4167" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4161" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4162" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4163" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4164" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4165" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4166" class="MJXp-mo MJXp-script">′</span></span></span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4168" class="MJXp-mo">⟩</span></span></span><span tabindex="0" id="MathJax-Element-679-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-679" type="math/tex">\langle\overline{a_{2'}}\rangle</script></span> has a cyclic complement <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4169" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4170" class="MJXp-mo">⟨</span><span id="MJXp-Span-4171" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4178" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4172" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4173" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4174" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4175" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4176" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4177" class="MJXp-mo MJXp-script">′</span></span></span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4179" class="MJXp-mo">⟩</span></span></span><span tabindex="0" id="MathJax-Element-680-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-680" type="math/tex">\langle\overline{b_{2'}}\rangle</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4180" class="MJXp-math"><span id="MJXp-Span-4181" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4182" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4183" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4184" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4185" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4186" class="MJXp-mo MJXp-script">′</span></span></span></span><span id="MJXp-Span-4187" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4188" class="MJXp-mo">/</span></span><span id="MJXp-Span-4189" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-681-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-681" type="math/tex">E_{2'}/K</script></span>. Using the description of the primitive central idempotents and the Wedderburn components of a semisimple finite group algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4190" class="MJXp-math"><span id="MJXp-Span-4191" class="MJXp-mi MJXp-italic">F</span><span id="MJXp-Span-4192" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-682-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-682" type="math/tex">F G</script></span> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.17</span></a>), a complete set of orthogonal primitive idempotents of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4193" class="MJXp-math"><span id="MJXp-Span-4194" class="MJXp-mrow"><span id="MJXp-Span-4195" class="MJXp-mi undefined">F</span></span><span id="MJXp-Span-4196" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-4197" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4198" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4199" class="MJXp-mi MJXp-italic MJXp-script">C</span></span></span></span><span tabindex="0" id="MathJax-Element-683-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-683" type="math/tex">\mathbb F Ge_C</script></span> is described (see <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBOV">[OG11]</a>) as the set of conjugates of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4200" class="MJXp-math"><span id="MJXp-Span-4201" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4202" class="MJXp-mi MJXp-italic">β</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4203" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4204" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4205" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4206" class="MJXp-mi MJXp-italic MJXp-script">C</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4207" class="MJXp-mo">=</span><span id="MJXp-Span-4208" class="MJXp-mrow"><span id="MJXp-Span-4209" class="MJXp-munderover"><span><span class="MJXp-over"><span class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-4216" class="MJXp-mo">~</span></span><span class=""><span id="MJXp-Span-4210" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4211" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4212" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4213" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4214" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4215" class="MJXp-mo MJXp-script">′</span></span></span></span></span></span></span></span></span><span id="MJXp-Span-4217" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4218" class="MJXp-mi MJXp-italic">β</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4219" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4220" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4221" class="MJXp-mi MJXp-italic">ε</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4222" class="MJXp-mi MJXp-italic MJXp-script">C</span></span></span></span><span tabindex="0" id="MathJax-Element-684-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-684" type="math/tex">\beta_{e_C}=\widetilde{b_{2'}}\beta_2\varepsilon_C</script></span> by the elements of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4223" class="MJXp-math"><span id="MJXp-Span-4224" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4225" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4226" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4227" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4228" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4229" class="MJXp-mi MJXp-italic MJXp-script">C</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4230" class="MJXp-mo">=</span><span id="MJXp-Span-4231" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4232" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4233" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4234" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4235" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4236" class="MJXp-mo MJXp-script">′</span></span></span></span><span id="MJXp-Span-4237" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4238" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4239" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4240" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4241" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4242" class="MJXp-mi MJXp-italic MJXp-script">E</span></span></span></span><span tabindex="0" id="MathJax-Element-685-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-685" type="math/tex">T_{e_C}=T_{2'}T_2T_E</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4243" class="MJXp-math"><span id="MJXp-Span-4244" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4245" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4246" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4247" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4248" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4249" class="MJXp-mo MJXp-script">′</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4250" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4251" class="MJXp-mo">{</span><span id="MJXp-Span-4252" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4253" class="MJXp-mo">,</span><span id="MJXp-Span-4254" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4255" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4256" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4257" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4258" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4259" class="MJXp-mo MJXp-script">′</span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4260" class="MJXp-mo">,</span><span id="MJXp-Span-4261" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4262" class="MJXp-mi MJXp-italic">a</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4267" class="MJXp-mn">2</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-4263" class="MJXp-mrow"><span id="MJXp-Span-4264" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4265" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4266" class="MJXp-mo MJXp-script">′</span></span></span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4268" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4269" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4270" class="MJXp-mo">,</span><span id="MJXp-Span-4271" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4272" class="MJXp-mi MJXp-italic">a</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.59em;"><span id="MJXp-Span-4277" class="MJXp-mrow"><span id="MJXp-Span-4278" class="MJXp-mo">[</span><span id="MJXp-Span-4279" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4280" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4281" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4282" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4283" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4284" class="MJXp-mo MJXp-script">′</span></span></span></span><span id="MJXp-Span-4285" class="MJXp-mo">:</span><span id="MJXp-Span-4286" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4287" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4288" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4289" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4290" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4291" class="MJXp-mo MJXp-script">′</span></span></span></span><span id="MJXp-Span-4292" class="MJXp-mo">]</span><span id="MJXp-Span-4293" class="MJXp-mo">−</span><span id="MJXp-Span-4294" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-4273" class="MJXp-mrow"><span id="MJXp-Span-4274" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-4275" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4276" class="MJXp-mo MJXp-script">′</span></span></span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4295" class="MJXp-mo">}</span></span></span><span tabindex="0" id="MathJax-Element-686-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-686" type="math/tex">T_{2'}=\{1,a_{2'},a_{2'}^2,\dots,a_{2'}^{[E_{2'}:H_{2'}]-1}\}</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4296" class="MJXp-math"><span id="MJXp-Span-4297" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4298" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4299" class="MJXp-mi MJXp-italic MJXp-script">E</span></span></span></span><span tabindex="0" id="MathJax-Element-687-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-687" type="math/tex">T_E</script></span> denotes a right transversal of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4300" class="MJXp-math"><span id="MJXp-Span-4301" class="MJXp-mi MJXp-italic">E</span></span></span><span tabindex="0" id="MathJax-Element-688-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-688" type="math/tex">E</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4302" class="MJXp-math"><span id="MJXp-Span-4303" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-689-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-689" type="math/tex">G</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4304" class="MJXp-math"><span id="MJXp-Span-4305" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4306" class="MJXp-mi MJXp-italic">β</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4307" class="MJXp-mn MJXp-script">2</span></span></span></span><span tabindex="0" id="MathJax-Element-690-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-690" type="math/tex">\beta_2</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4308" class="MJXp-math"><span id="MJXp-Span-4309" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4310" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4311" class="MJXp-mn MJXp-script">2</span></span></span></span><span tabindex="0" id="MathJax-Element-691-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-691" type="math/tex">T_2</script></span> are given according to the cases below.</p>
-
-<ol>
-<li><p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4312" class="MJXp-math"><span id="MJXp-Span-4313" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4314" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4315" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4316" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4317" class="MJXp-mo">/</span></span><span id="MJXp-Span-4318" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-692-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-692" type="math/tex">H_2/K</script></span> has a complement <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4319" class="MJXp-math"><span id="MJXp-Span-4320" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4321" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4322" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4323" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4324" class="MJXp-mo">/</span></span><span id="MJXp-Span-4325" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-693-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-693" type="math/tex">M_2/K</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4326" class="MJXp-math"><span id="MJXp-Span-4327" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4328" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4329" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4330" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4331" class="MJXp-mo">/</span></span><span id="MJXp-Span-4332" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-694-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-694" type="math/tex">E_2/K</script></span> then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4333" class="MJXp-math"><span id="MJXp-Span-4334" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4335" class="MJXp-mi MJXp-italic">β</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4336" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4337" class="MJXp-mo">=</span><span id="MJXp-Span-4338" class="MJXp-mrow"><span id="MJXp-Span-4339" class="MJXp-munderover"><span><span class="MJXp-over"><span class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-4343" class="MJXp-mo">~</span></span><span class=""><span id="MJXp-Span-4340" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4341" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4342" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-695-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-695" type="math/tex">\beta_2=\widetilde{M_2}</script></span>. Moreover, if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4344" class="MJXp-math"><span id="MJXp-Span-4345" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4346" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4347" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4348" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4349" class="MJXp-mo">/</span></span><span id="MJXp-Span-4350" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-696-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-696" type="math/tex">M_2/K</script></span> is cyclic, then there exists <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4351" class="MJXp-math"><span id="MJXp-Span-4352" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4353" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4354" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4355" class="MJXp-mo">∈</span><span id="MJXp-Span-4356" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4357" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4358" class="MJXp-mn MJXp-script">2</span></span></span></span><span tabindex="0" id="MathJax-Element-697-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-697" type="math/tex">b_2\in E_2</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4359" class="MJXp-math"><span id="MJXp-Span-4360" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4361" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4362" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4363" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4364" class="MJXp-mo">/</span></span><span id="MJXp-Span-4365" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-698-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-698" type="math/tex">E_2/K</script></span> is given by the following presentation</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4366" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4367" class="MJXp-mo">⟨</span><span id="MJXp-Span-4368" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4372" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4369" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4370" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4371" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4373" class="MJXp-mo">,</span><span id="MJXp-Span-4374" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4378" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4375" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4376" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4377" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4379" class="MJXp-mo">∣</span><span id="MJXp-Span-4380" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4384" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4381" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4382" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4383" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4385" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4386" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4387" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4388" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4389" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4390" class="MJXp-mi MJXp-italic MJXp-script">n</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4391" class="MJXp-mo">=</span><span id="MJXp-Span-4392" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4396" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4393" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4394" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4395" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4397" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4398" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4399" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4400" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4401" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4402" class="MJXp-mi MJXp-italic MJXp-script">k</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4403" class="MJXp-mo">=</span><span id="MJXp-Span-4404" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4405" class="MJXp-mo">,</span><span id="MJXp-Span-4406" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4410" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4407" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4408" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4409" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4411" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4412" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4413" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4414" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4418" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4415" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4416" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4417" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4419" class="MJXp-mo">=</span><span id="MJXp-Span-4420" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4424" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4421" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4422" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4423" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4425" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4426" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4427" class="MJXp-mi MJXp-italic MJXp-script">r</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4428" class="MJXp-mo">⟩</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4429" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processing"><span tabindex="0" id="MathJax-Element-699-Frame" class="MathJax"></span></div><script id="MathJax-Element-699" type="math/tex; mode=display">\langle \overline{a_2},\overline{b_2}\mid \overline{a_2}\hspace{1pt}^{2^n}=\overline{b_2}\hspace{1pt}^{2^k}=1,
- \overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r \rangle,</script></p>
-
-<p>and if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4430" class="MJXp-math"><span id="MJXp-Span-4431" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4432" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4433" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4434" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4435" class="MJXp-mo">/</span></span><span id="MJXp-Span-4436" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-700-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-700" type="math/tex">M_2/K</script></span> is not cyclic, then there exist <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4437" class="MJXp-math"><span id="MJXp-Span-4438" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4439" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4440" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4441" class="MJXp-mo">,</span><span id="MJXp-Span-4442" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4443" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4444" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4445" class="MJXp-mo">∈</span><span id="MJXp-Span-4446" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4447" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4448" class="MJXp-mn MJXp-script">2</span></span></span></span><span tabindex="0" id="MathJax-Element-701-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-701" type="math/tex">b_2,c_2\in E_2</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4449" class="MJXp-math"><span id="MJXp-Span-4450" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4451" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4452" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4453" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4454" class="MJXp-mo">/</span></span><span id="MJXp-Span-4455" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-702-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-702" type="math/tex">E_2/K</script></span> is given by the following presentation</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4456" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4457" class="MJXp-mo">⟨</span><span id="MJXp-Span-4458" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4462" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4459" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4460" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4461" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4463" class="MJXp-mo">,</span><span id="MJXp-Span-4464" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4468" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4465" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4466" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4467" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4469" class="MJXp-mo">,</span><span id="MJXp-Span-4470" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4474" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4471" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4472" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4473" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4475" class="MJXp-mo">∣</span><span id="MJXp-Span-4476" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4480" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4477" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4478" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4479" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4481" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4482" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4483" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4484" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4485" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4486" class="MJXp-mi MJXp-italic MJXp-script">n</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4487" class="MJXp-mo">=</span><span id="MJXp-Span-4488" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4492" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4489" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4490" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4491" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4493" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4494" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4495" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4496" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4497" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4498" class="MJXp-mi MJXp-italic MJXp-script">k</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4499" class="MJXp-mo">=</span><span id="MJXp-Span-4500" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4504" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4501" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4502" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4503" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4505" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4506" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4507" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4508" class="MJXp-mo">=</span><span id="MJXp-Span-4509" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4510" class="MJXp-mo">,</span><span id="MJXp-Span-4511" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4515" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4512" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4513" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4514" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4516" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4517" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4518" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4519" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4523" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4520" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4521" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4522" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4524" class="MJXp-mo">=</span><span id="MJXp-Span-4525" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4529" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4526" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4527" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4528" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4530" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4531" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4532" class="MJXp-mi MJXp-italic MJXp-script">r</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4533" class="MJXp-mo">,</span><span id="MJXp-Span-4534" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4538" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4535" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4536" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4537" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4539" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4540" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4541" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4542" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4546" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4543" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4544" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4545" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4547" class="MJXp-mo">=</span><span id="MJXp-Span-4548" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4552" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4549" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4550" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4551" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4553" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4554" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4555" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4556" class="MJXp-mo">−</span><span id="MJXp-Span-4557" class="MJXp-mn">1</span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4558" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4559" class="MJXp-mo">[</span><span id="MJXp-Span-4560" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4564" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4561" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4562" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4563" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4565" class="MJXp-mo">,</span><span id="MJXp-Span-4566" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4570" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4567" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4568" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4569" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4571" class="MJXp-mo">]</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4572" class="MJXp-mo">=</span><span id="MJXp-Span-4573" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4574" class="MJXp-mo">⟩</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4575" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processing"><span tabindex="0" id="MathJax-Element-703-Frame" class="MathJax"></span></div><script id="MathJax-Element-703" type="math/tex; mode=display">\langle \overline{a_2},\overline{b_2},\overline{c_2}\mid \overline{a_2}\hspace{1pt}^{2^n}=
-\overline{b_2}\hspace{1pt}^{2^k}=\overline{c_2}\hspace{1pt}^2=1,
-\overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r,
-\overline{a_2}\hspace{1pt}^{\overline{c_2}}=\overline{a_2}\hspace{1pt}^{-1}, [\overline{b_2},\overline{c_2}]=1 \rangle,</script></p>
-
-<p>with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4576" class="MJXp-math"><span id="MJXp-Span-4577" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4578" class="MJXp-mo">≡</span><span id="MJXp-Span-4579" class="MJXp-mn">1</span><span id="MJXp-Span-4580" class="MJXp-mrow"><span style="width: 0.167em; height: 0em;" id="MJXp-Span-4581" class="MJXp-mspace"></span><span id="MJXp-Span-4582" class="MJXp-mi">m</span><span id="MJXp-Span-4583" class="MJXp-mi">o</span><span id="MJXp-Span-4584" class="MJXp-mi">d</span><span style="width: 0.167em; height: 0em;" id="MJXp-Span-4585" class="MJXp-mspace"></span></span><span id="MJXp-Span-4586" class="MJXp-mn">4</span></span></span><span tabindex="0" id="MathJax-Element-704-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-704" type="math/tex">r \equiv 1 {\rm \,mod\,} 4</script></span> (or equivalently <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4587" class="MJXp-math"><span id="MJXp-Span-4588" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4592" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4589" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4590" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4591" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4593" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4594" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4595" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4596" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4597" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4598" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4599" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-4600" class="MJXp-mo">−</span><span id="MJXp-Span-4601" class="MJXp-mn">2</span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-705-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-705" type="math/tex">\overline{a_2}\hspace{1pt}^{2^{n-2}}</script></span> is central in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4602" class="MJXp-math"><span id="MJXp-Span-4603" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4604" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4605" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4606" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4607" class="MJXp-mo">/</span></span><span id="MJXp-Span-4608" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-706-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-706" type="math/tex">E_2/K</script></span>). Then</p>
-
-<ol>
-<li><p><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4609" class="MJXp-math"><span id="MJXp-Span-4610" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4611" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4612" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4613" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4614" class="MJXp-mo">{</span><span id="MJXp-Span-4615" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4616" class="MJXp-mo">,</span><span id="MJXp-Span-4617" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4618" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4619" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4620" class="MJXp-mo">,</span><span id="MJXp-Span-4621" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4622" class="MJXp-mi MJXp-italic">a</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4624" class="MJXp-mn">2</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-4623" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4625" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4626" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4627" class="MJXp-mo">,</span><span id="MJXp-Span-4628" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4629" class="MJXp-mi MJXp-italic">a</span><span style="height: 2.116em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4631" class="MJXp-mrow"><span id="MJXp-Span-4632" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4633" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4634" class="MJXp-mi MJXp-italic MJXp-script">k</span></span><span id="MJXp-Span-4635" class="MJXp-mo">−</span><span id="MJXp-Span-4636" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -1.17em;"><span id="MJXp-Span-4630" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4637" class="MJXp-mo">}</span></span></span><span tabindex="0" id="MathJax-Element-707-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-707" type="math/tex">T_2=\{1,a_2,a_2^2,\dots, a_2^{2^k-1}\}</script></span>, if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4638" class="MJXp-math"><span id="MJXp-Span-4639" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4643" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4640" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4641" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4642" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4644" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4645" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4646" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4647" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4648" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4649" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4650" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-4651" class="MJXp-mo">−</span><span id="MJXp-Span-4652" class="MJXp-mn">2</span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-708-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-708" type="math/tex">\overline{a_2}\hspace{1pt}^{2^{n-2}}</script></span> is central in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4653" class="MJXp-math"><span id="MJXp-Span-4654" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4655" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4656" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4657" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4658" class="MJXp-mo">/</span></span><span id="MJXp-Span-4659" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-709-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-709" type="math/tex">E_2/K</script></span> (unless <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4660" class="MJXp-math"><span id="MJXp-Span-4661" class="MJXp-mi MJXp-italic">n</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4662" class="MJXp-mo">≤</span><span id="MJXp-Span-4663" class="MJXp-mn">1</span></span></span><span tabindex="0" id="MathJax-Element-710-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-710" type="math/tex">n\leq 1</script></span>) and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4664" class="MJXp-math"><span id="MJXp-Span-4665" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4666" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4667" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4668" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4669" class="MJXp-mo">/</span></span><span id="MJXp-Span-4670" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-711-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-711" type="math/tex">M_2/K</script></span> is cyclic; and</p>
-
-</li>
-<li><p><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4671" class="MJXp-math"><span id="MJXp-Span-4672" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4673" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4674" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4675" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4676" class="MJXp-mo">{</span><span id="MJXp-Span-4677" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4678" class="MJXp-mo">,</span><span id="MJXp-Span-4679" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4680" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4681" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4682" class="MJXp-mo">,</span><span id="MJXp-Span-4683" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4684" class="MJXp-mi MJXp-italic">a</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4686" class="MJXp-mn">2</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-4685" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4687" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4688" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4689" class="MJXp-mo">,</span><span id="MJXp-Span-4690" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4691" class="MJXp-mi MJXp-italic">a</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4693" class="MJXp-mrow"><span id="MJXp-Span-4694" class="MJXp-mi MJXp-italic">d</span><span id="MJXp-Span-4695" class="MJXp-mrow"><span id="MJXp-Span-4696" class="MJXp-mo">/</span></span><span id="MJXp-Span-4697" class="MJXp-mn">2</span><span id="MJXp-Span-4698" class="MJXp-mo">−</span><span id="MJXp-Span-4699" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-4692" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4700" class="MJXp-mo">,</span><span id="MJXp-Span-4701" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4702" class="MJXp-mi MJXp-italic">a</span><span style="height: 2.116em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4704" class="MJXp-mrow"><span id="MJXp-Span-4705" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4706" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4707" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4708" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-4709" class="MJXp-mo">−</span><span id="MJXp-Span-4710" class="MJXp-mn">2</span></span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -1.17em;"><span id="MJXp-Span-4703" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4711" class="MJXp-mo">,</span><span id="MJXp-Span-4712" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4713" class="MJXp-mi MJXp-italic">a</span><span style="height: 2.116em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4715" class="MJXp-mrow"><span id="MJXp-Span-4716" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4717" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4718" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4719" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-4720" class="MJXp-mo">−</span><span id="MJXp-Span-4721" class="MJXp-mn">2</span></span></span><span id="MJXp-Span-4722" class="MJXp-mo">+</span><span id="MJXp-Span-4723" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -1.17em;"><span id="MJXp-Span-4714" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4724" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4725" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4726" class="MJXp-mo">,</span><span id="MJXp-Span-4727" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4728" class="MJXp-mi MJXp-italic">a</span><span style="height: 2.116em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4730" class="MJXp-mrow"><span id="MJXp-Span-4731" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4732" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4733" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4734" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-4735" class="MJXp-mo">−</span><span id="MJXp-Span-4736" class="MJXp-mn">2</span></span></span><span id="MJXp-Span-4737" class="MJXp-mo">+</span><span id="MJXp-Span-4738" class="MJXp-mi MJXp-italic">d</span><span id="MJXp-Span-4739" class="MJXp-mrow"><span id="MJXp-Span-4740" class="MJXp-mo">/</span></span><span id="MJXp-Span-4741" class="MJXp-mn">2</span><span id="MJXp-Span-4742" class="MJXp-mo">−</span><span id="MJXp-Span-4743" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -1.17em;"><span id="MJXp-Span-4729" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4744" class="MJXp-mo">}</span></span></span><span tabindex="0" id="MathJax-Element-712-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-712" type="math/tex">T_2=\{1,a_2,a_2^2,\dots,a_2^{d/2-1},a_2^{2^{n-2}},a_2^{2^{n-2}+1},\dots,a_2^{2^{n-2}+d/2-1}\}</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4745" class="MJXp-math"><span id="MJXp-Span-4746" class="MJXp-mi MJXp-italic">d</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4747" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4748" class="MJXp-mo">[</span><span id="MJXp-Span-4749" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4750" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4751" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-4752" class="MJXp-mo">:</span><span id="MJXp-Span-4753" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4754" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4755" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4756" class="MJXp-mo">]</span></span></span><span tabindex="0" id="MathJax-Element-713-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-713" type="math/tex">d=[E_2:H_2]</script></span>, otherwise.</p>
-
-</li>
-</ol>
-</li>
-<li><p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4757" class="MJXp-math"><span id="MJXp-Span-4758" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4759" class="MJXp-mi MJXp-italic">H</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4760" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4761" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4762" class="MJXp-mo">/</span></span><span id="MJXp-Span-4763" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-714-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-714" type="math/tex">H_2/K</script></span> has no complement in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4764" class="MJXp-math"><span id="MJXp-Span-4765" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4766" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4767" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4768" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4769" class="MJXp-mo">/</span></span><span id="MJXp-Span-4770" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-715-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-715" type="math/tex">E_2/K</script></span>, then there exist <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4771" class="MJXp-math"><span id="MJXp-Span-4772" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4773" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4774" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4775" class="MJXp-mo">,</span><span id="MJXp-Span-4776" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4777" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4778" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4779" class="MJXp-mo">∈</span><span id="MJXp-Span-4780" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4781" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4782" class="MJXp-mn MJXp-script">2</span></span></span></span><span tabindex="0" id="MathJax-Element-716-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-716" type="math/tex">b_2,c_2\in E_2</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4783" class="MJXp-math"><span id="MJXp-Span-4784" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4785" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4786" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-4787" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-4788" class="MJXp-mo">/</span></span><span id="MJXp-Span-4789" class="MJXp-mi MJXp-italic">K</span></span></span><span tabindex="0" id="MathJax-Element-717-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-717" type="math/tex">E_2/K</script></span> is given by the following presentation</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4790" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4791" class="MJXp-mo">⟨</span><span id="MJXp-Span-4792" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4796" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4793" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4794" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4795" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4797" class="MJXp-mo">,</span><span id="MJXp-Span-4798" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4802" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4799" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4800" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4801" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4803" class="MJXp-mo">,</span><span id="MJXp-Span-4804" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4808" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4805" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4806" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4807" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4809" class="MJXp-mo">∣</span><span id="MJXp-Span-4810" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4814" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4811" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4812" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4813" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4815" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4816" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4817" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4818" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4819" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4820" class="MJXp-mi MJXp-italic MJXp-script">n</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4821" class="MJXp-mo">=</span><span id="MJXp-Span-4822" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4826" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4823" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4824" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4825" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4827" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4828" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4829" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4830" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4831" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4832" class="MJXp-mi MJXp-italic MJXp-script">k</span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4833" class="MJXp-mo">=</span><span id="MJXp-Span-4834" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4835" class="MJXp-mo">,</span><span id="MJXp-Span-4836" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4840" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4837" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4838" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4839" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4841" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4842" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4843" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4844" class="MJXp-mo">=</span><span id="MJXp-Span-4845" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4849" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4846" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4847" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4848" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4850" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4851" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4852" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4853" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4854" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4855" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4856" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-4857" class="MJXp-mo">−</span><span id="MJXp-Span-4858" class="MJXp-mn">1</span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4859" class="MJXp-mo">,</span><span id="MJXp-Span-4860" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4864" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4861" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4862" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4863" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4865" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4866" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4867" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4868" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4872" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4869" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4870" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4871" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4873" class="MJXp-mo">=</span><span id="MJXp-Span-4874" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4878" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4875" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4876" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4877" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4879" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4880" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4881" class="MJXp-mi MJXp-italic MJXp-script">r</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4882" class="MJXp-mo">,</span><span style="width: 0em; height: 0em;" id="MJXp-Span-4883" class="MJXp-mspace"></span><span id="MJXp-Span-4884" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4888" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4885" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4886" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4887" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4889" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4890" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4891" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4892" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4896" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4893" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4894" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4895" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4897" class="MJXp-mo">=</span><span id="MJXp-Span-4898" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4902" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4899" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4900" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4901" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span id="MJXp-Span-4903" class="MJXp-msubsup"><span style="width: 0.1em; height: 0em; margin-right: 0.05em;" id="MJXp-Span-4904" class="MJXp-mspace"></span><span style="vertical-align: 0.4em;" id="MJXp-Span-4905" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4906" class="MJXp-mo">−</span><span id="MJXp-Span-4907" class="MJXp-mn">1</span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4908" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4909" class="MJXp-mo">[</span><span id="MJXp-Span-4910" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4914" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4911" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4912" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4913" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4915" class="MJXp-mo">,</span><span id="MJXp-Span-4916" class="MJXp-munderover"><span><span class="MJXp-over"><span class=" MJXp-script"><span style="margin-right: 0px; margin-left: 0px;" id="MJXp-Span-4920" class="MJXp-mo">¯</span></span><span class=""><span id="MJXp-Span-4917" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4918" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4919" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4921" class="MJXp-mo">]</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4922" class="MJXp-mo">=</span><span id="MJXp-Span-4923" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4924" class="MJXp-mo">⟩</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4925" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processing"><span tabindex="0" id="MathJax-Element-718-Frame" class="MathJax"></span></div><script id="MathJax-Element-718" type="math/tex; mode=display"> \langle \overline{a_2},\overline{b_2},\overline{c_2}\mid \overline{a_2}\hspace{1pt}^{2^n}=
-\overline{b_2}\hspace{1pt}^{2^k}=1, \overline{c_2}\hspace{1pt}^2=\overline{a_2}\hspace{1pt}^{2^{n-1}},
-\overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r,\\
-\overline{a_2}\hspace{1pt}^{\overline{c_2}}=\overline{a_2}\hspace{1pt}^{-1},[\overline{b_2},\overline{c_2}]=1 \rangle, </script></p>
-
-<p>with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4926" class="MJXp-math"><span id="MJXp-Span-4927" class="MJXp-mi MJXp-italic">r</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4928" class="MJXp-mo">≡</span><span id="MJXp-Span-4929" class="MJXp-mn">1</span><span id="MJXp-Span-4930" class="MJXp-mrow"><span style="width: 0.167em; height: 0em;" id="MJXp-Span-4931" class="MJXp-mspace"></span><span id="MJXp-Span-4932" class="MJXp-mi">m</span><span id="MJXp-Span-4933" class="MJXp-mi">o</span><span id="MJXp-Span-4934" class="MJXp-mi">d</span><span style="width: 0.167em; height: 0em;" id="MJXp-Span-4935" class="MJXp-mspace"></span></span><span id="MJXp-Span-4936" class="MJXp-mn">4</span></span></span><span tabindex="0" id="MathJax-Element-719-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-719" type="math/tex">r\equiv 1 {\rm \,mod\,} 4</script></span>. In this case, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4937" class="MJXp-math"><span id="MJXp-Span-4938" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4939" class="MJXp-mi MJXp-italic">β</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4940" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4941" class="MJXp-mo">=</span><span id="MJXp-Span-4942" class="MJXp-mrow"><span id="MJXp-Span-4943" class="MJXp-munderover"><span><span class="MJXp-over"><span class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-4947" class="MJXp-mo">~</span></span><span class=""><span id="MJXp-Span-4944" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4945" class="MJXp-mi MJXp-italic">b</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4946" class="MJXp-mn MJXp-script">2</span></span></span></span></span></span></span><span style="vertical-align: 0.25em;" id="MJXp-Span-4948" class="MJXp-mfrac"><span class="MJXp-box MJXp-script"><span id="MJXp-Span-4949" class="MJXp-mn">1</span><span id="MJXp-Span-4950" class="MJXp-mo">+</span><span id="MJXp-Span-4951" class="MJXp-mi MJXp-italic">x</span><span id="MJXp-Span-4952" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4953" class="MJXp-mi MJXp-italic">a</span><span style="height: 2.116em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4955" class="MJXp-mrow"><span id="MJXp-Span-4956" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4957" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4958" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4959" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-4960" class="MJXp-mo">−</span><span id="MJXp-Span-4961" class="MJXp-mn">2</span></span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -1.17em;"><span id="MJXp-Span-4954" class="MJXp-mn">2</span></span></span></span></span></span><span id="MJXp-Span-4962" class="MJXp-mo">+</span><span id="MJXp-Span-4963" class="MJXp-mi MJXp-italic">y</span><span id="MJXp-Span-4964" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4965" class="MJXp-mi MJXp-italic">a</span><span style="height: 2.116em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4967" class="MJXp-mrow"><span id="MJXp-Span-4968" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4969" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-4970" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-4971" class="MJXp-mi MJXp-italic">n</span><span id="MJXp-Span-4972" class="MJXp-mo">−</span><span id="MJXp-Span-4973" class="MJXp-mn">2</span></span></span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -1.17em;"><span id="MJXp-Span-4966" class="MJXp-mn">2</span></span></span></span></span></span><span id="MJXp-Span-4974" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4975" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4976" class="MJXp-mn MJXp-script">2</span></span></span><span style="margin-top: -0.9em;" class="MJXp-box"><span class="MJXp-denom"><span><span style="height: 1em; border-top: medium none; border-bottom: 1px solid; margin: 0.1em 0px;" class="MJXp-rule"></span></span><span><span class="MJXp-box MJXp-script"><span id="MJXp-Span-4977" class="MJXp-mn">2</span></span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-720-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-720" type="math/tex">\beta_2=\widetilde{b_2}\frac{1+xa_2^{2^{n-2}}+ya_2^{2^{n-2}}c_2}{2}</script></span> and</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-4978" class="MJXp-math MJXp-display"><span id="MJXp-Span-4979" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4980" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4981" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-4982" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4983" class="MJXp-mo">{</span><span id="MJXp-Span-4984" class="MJXp-mn">1</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4985" class="MJXp-mo">,</span><span id="MJXp-Span-4986" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4987" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-4988" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4989" class="MJXp-mo">,</span><span id="MJXp-Span-4990" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4991" class="MJXp-mi MJXp-italic">a</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-4993" class="MJXp-mn">2</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-4992" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4994" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-4995" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-4996" class="MJXp-mo">,</span><span id="MJXp-Span-4997" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-4998" class="MJXp-mi MJXp-italic">a</span><span style="height: 2.116em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-5000" class="MJXp-mrow"><span id="MJXp-Span-5001" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5002" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5003" class="MJXp-mi MJXp-italic MJXp-script">k</span></span><span id="MJXp-Span-5004" class="MJXp-mo">−</span><span id="MJXp-Span-5005" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -1.17em;"><span id="MJXp-Span-4999" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5006" class="MJXp-mo">,</span><span id="MJXp-Span-5007" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5008" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5009" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5010" class="MJXp-mo">,</span><span id="MJXp-Span-5011" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5012" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5013" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-5014" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5015" class="MJXp-mi MJXp-italic">a</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5016" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5017" class="MJXp-mo">,</span><span id="MJXp-Span-5018" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5019" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5020" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-5021" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5022" class="MJXp-mi MJXp-italic">a</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-5024" class="MJXp-mn">2</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-5023" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5025" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5026" class="MJXp-mo">…</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5027" class="MJXp-mo">,</span><span id="MJXp-Span-5028" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5029" class="MJXp-mi MJXp-italic">c</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5030" class="MJXp-mn MJXp-script">2</span></span><span id="MJXp-Span-5031" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5032" class="MJXp-mi MJXp-italic">a</span><span style="height: 2.116em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-5034" class="MJXp-mrow"><span id="MJXp-Span-5035" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5036" class="MJXp-mn">2</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5037" class="MJXp-mi MJXp-italic MJXp-script">k</span></span><span id="MJXp-Span-5038" class="MJXp-mo">−</span><span id="MJXp-Span-5039" class="MJXp-mn">1</span></span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -1.17em;"><span id="MJXp-Span-5033" class="MJXp-mn">2</span></span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5040" class="MJXp-mo">}</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5041" class="MJXp-mo">,</span></span></span><div class="MathJax_Display MathJax_Processing"><span tabindex="0" id="MathJax-Element-721-Frame" class="MathJax"></span></div><script id="MathJax-Element-721" type="math/tex; mode=display">T_2=\{1,a_2,a_2^2,\dots, a_2^{2^k-1},c_2,c_2a_2,c_2a_2^2,\dots,c_2a_2^{2^k-1}\},</script></p>
-
-<p>with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5042" class="MJXp-math"><span id="MJXp-Span-5043" class="MJXp-mi MJXp-italic">x</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5044" class="MJXp-mo">,</span><span id="MJXp-Span-5045" class="MJXp-mi MJXp-italic">y</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5046" class="MJXp-mo">∈</span><span id="MJXp-Span-5047" class="MJXp-mrow"><span id="MJXp-Span-5048" class="MJXp-mi undefined">F</span></span></span></span><span tabindex="0" id="MathJax-Element-722-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-722" type="math/tex">x,y\in\mathbb F</script></span>, satisfying <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5049" class="MJXp-math"><span id="MJXp-Span-5050" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5051" class="MJXp-mi MJXp-italic">x</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5052" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-5053" class="MJXp-mo">+</span><span id="MJXp-Span-5054" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5055" class="MJXp-mi MJXp-italic">y</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5056" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5057" class="MJXp-mo">=</span><span style="margin-left: 0.267em; margin-right: 0.267em;" id="MJXp-Span-5058" class="MJXp-mo">−</span><span id="MJXp-Span-5059" class="MJXp-mn">1</span></span></span><span tabindex="0" id="MathJax-Element-723-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-723" type="math/tex">x^2+y^2=-1</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5060" class="MJXp-math"><span id="MJXp-Span-5061" class="MJXp-mi MJXp-italic">y</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5062" class="MJXp-mo">≠</span><span id="MJXp-Span-5063" class="MJXp-mn">0</span></span></span><span tabindex="0" id="MathJax-Element-724-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-724" type="math/tex">y\neq 0</script></span>.</p>
-
-</li>
-</ol>
-<p>When <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5064" class="MJXp-math"><span id="MJXp-Span-5065" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-725-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-725" type="math/tex">G</script></span> is not nilpotent, we can still use the following description in some specific cases. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5066" class="MJXp-math"><span id="MJXp-Span-5067" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-726-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-726" type="math/tex">G</script></span> be a finite group and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5068" class="MJXp-math"><span id="MJXp-Span-5069" class="MJXp-mrow"><span id="MJXp-Span-5070" class="MJXp-mi undefined">F</span></span></span></span><span tabindex="0" id="MathJax-Element-727-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-727" type="math/tex">\mathbb F</script></span> a finite field of order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5071" class="MJXp-math"><span id="MJXp-Span-5072" class="MJXp-mi MJXp-italic">s</span></span></span><span tabindex="0" id="MathJax-Element-728-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-728" type="math/tex">s</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5073" class="MJXp-math"><span id="MJXp-Span-5074" class="MJXp-mi MJXp-italic">s</span></span></span><span tabindex="0" id="MathJax-Element-729-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-729" type="math/tex">s</script></span> is coprime to the order of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5075" class="MJXp-math"><span id="MJXp-Span-5076" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-730-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-730" type="math/tex">G</script></span>. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5077" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5078" class="MJXp-mo">(</span><span id="MJXp-Span-5079" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5080" class="MJXp-mo">,</span><span id="MJXp-Span-5081" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5082" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-731-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-731" type="math/tex">(H,K)</script></span> be a strong Shoda pair of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5083" class="MJXp-math"><span id="MJXp-Span-5084" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-732-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-732" type="math/tex">G</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5085" class="MJXp-math"><span id="MJXp-Span-5086" class="MJXp-mi MJXp-italic">τ</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5087" class="MJXp-mo">(</span><span id="MJXp-Span-5088" class="MJXp-mi MJXp-italic">g</span><span id="MJXp-Span-5089" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5090" class="MJXp-mo">,</span><span id="MJXp-Span-5091" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-5092" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5093" class="MJXp-mo MJXp-script">′</span></span><span id="MJXp-Span-5094" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5095" class="MJXp-mo">)</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5096" class="MJXp-mo">=</span><span id="MJXp-Span-5097" class="MJXp-mn">1</span></span></span><span tabindex="0" id="MathJax-Element-733-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-733" type="math/tex">\tau(gH,g'H)=1</script></span> for all <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5098" class="MJXp-math"><span id="MJXp-Span-5099" class="MJXp-mi MJXp-italic">g</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5100" class="MJXp-mo">,</span><span id="MJXp-Span-5101" class="MJXp-msup"><span style="margin-right: 0.05em;" id="MJXp-Span-5102" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5103" class="MJXp-mo MJXp-script">′</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5104" class="MJXp-mo">∈</span><span id="MJXp-Span-5105" class="MJXp-mi MJXp-italic">E</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5106" class="MJXp-mo">=</span><span id="MJXp-Span-5107" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5108" class="MJXp-mi MJXp-italic">E</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5109" class="MJXp-mi MJXp-italic MJXp-script">G</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5110" class="MJXp-mo">(</span><span id="MJXp-Span-5111" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-5112" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-5113" class="MJXp-mo">/</span></span><span id="MJXp-Span-5114" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5115" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-734-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-734" type="math/tex">g,g'\in E=E_G(H/K)</script></span>, and let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5116" class="MJXp-math"><span id="MJXp-Span-5117" class="MJXp-mi MJXp-italic">C</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5118" class="MJXp-mo">∈</span><span id="MJXp-Span-5119" class="MJXp-mrow"><span id="MJXp-Span-5120" class="MJXp-mi MJXp-cal">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5121" class="MJXp-mo">(</span><span id="MJXp-Span-5122" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-5123" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-5124" class="MJXp-mo">/</span></span><span id="MJXp-Span-5125" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5126" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-735-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-735" type="math/tex">C\in\mathcal{C}(H/K)</script></span>. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5127" class="MJXp-math"><span id="MJXp-Span-5128" class="MJXp-mi MJXp-italic">ε</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5129" class="MJXp-mo">=</span><span id="MJXp-Span-5130" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5131" class="MJXp-mi MJXp-italic">ε</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5132" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5133" class="MJXp-mo">(</span><span id="MJXp-Span-5134" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5135" class="MJXp-mo">,</span><span id="MJXp-Span-5136" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5137" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-736-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-736" type="math/tex">\varepsilon=\varepsilon_C(H,K)</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5138" class="MJXp-math"><span id="MJXp-Span-5139" class="MJXp-mi MJXp-italic">e</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5140" class="MJXp-mo">=</span><span id="MJXp-Span-5141" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5142" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5143" class="MJXp-mi MJXp-italic MJXp-script">C</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5144" class="MJXp-mo">(</span><span id="MJXp-Span-5145" class="MJXp-mi MJXp-italic">G</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5146" class="MJXp-mo">,</span><span id="MJXp-Span-5147" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5148" class="MJXp-mo">,</span><span id="MJXp-Span-5149" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5150" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-737-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-737" type="math/tex">e=e_C(G,H,K)</script></span> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.17</span></a>). Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5151" class="MJXp-math"><span id="MJXp-Span-5152" class="MJXp-mi MJXp-italic">w</span></span></span><span tabindex="0" id="MathJax-Element-738-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-738" type="math/tex">w</script></span> be a normal element of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5153" class="MJXp-math"><span id="MJXp-Span-5154" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5155" class="MJXp-mrow"><span id="MJXp-Span-5156" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5157" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5158" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5159" class="MJXp-mi MJXp-italic">s</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5160" class="MJXp-mi MJXp-italic MJXp-script">o</span></span></span></span><span id="MJXp-Span-5161" class="MJXp-mrow"><span style="margin-left: 0.111em; margin-right: 0.111em;" id="MJXp-Span-5162" class="MJXp-mo">/</span></span><span id="MJXp-Span-5163" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5164" class="MJXp-mrow"><span id="MJXp-Span-5165" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5166" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5167" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5168" class="MJXp-mi MJXp-italic">s</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5169" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5170" class="MJXp-mi MJXp-italic">o</span><span id="MJXp-Span-5171" class="MJXp-mrow"><span id="MJXp-Span-5172" class="MJXp-mo">/</span></span><span id="MJXp-Span-5173" class="MJXp-mo">[</span><span id="MJXp-Span-5174" class="MJXp-mi MJXp-italic">E</span><span id="MJXp-Span-5175" class="MJXp-mo">:</span><span id="MJXp-Span-5176" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-5177" class="MJXp-mo">]</span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-739-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-739" type="math/tex">\mathbb F_{s^o}/\mathbb F_{s^{o/[E:H]}}</script></span> (with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5178" class="MJXp-math"><span id="MJXp-Span-5179" class="MJXp-mi MJXp-italic">o</span></span></span><span tabindex="0" id="MathJax-Element-740-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-740" type="math/tex">o</script></span> the multiplicative order of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5180" class="MJXp-math"><span id="MJXp-Span-5181" class="MJXp-mi MJXp-italic">s</span></span></span><span tabindex="0" id="MathJax-Element-741-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-741" type="math/tex">s</script></span> modulo <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5182" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5183" class="MJXp-mo">[</span><span id="MJXp-Span-5184" class="MJXp-mi MJXp-italic">H</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-5185" class="MJXp-mo">:</span><span id="MJXp-Span-5186" class="MJXp-mi MJXp-italic">K</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5187" class="MJXp-mo">]</span></span></span><span tabindex="0" id="MathJax-Element-742-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-742" type="math/tex">[H:K]</script></span>) and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5188" class="MJXp-math"><span id="MJXp-Span-5189" class="MJXp-mi MJXp-italic">B</span></span></span><span tabindex="0" id="MathJax-Element-743-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-743" type="math/tex">B</script></span> the normal basis determined by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5190" class="MJXp-math"><span id="MJXp-Span-5191" class="MJXp-mi MJXp-italic">w</span></span></span><span tabindex="0" id="MathJax-Element-744-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-744" type="math/tex">w</script></span>. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5192" class="MJXp-math"><span id="MJXp-Span-5193" class="MJXp-mi MJXp-italic">ψ</span></span></span><span tabindex="0" id="MathJax-Element-745-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-745" type="math/tex">\psi</script></span> be the isomorphism between <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5194" class="MJXp-math"><span id="MJXp-Span-5195" class="MJXp-mrow"><span id="MJXp-Span-5196" class="MJXp-mi undefined">F</span></span><span id="MJXp-Span-5197" class="MJXp-mi MJXp-italic">E</span><span id="MJXp-Span-5198" class="MJXp-mi MJXp-italic">ε</span></span></span><span tabindex="0" id="MathJax-Element-746-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-746" type="math/tex">\mathbb F E \varepsilon</script></span> and the matrix algebra <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5199" class="MJXp-math"><span id="MJXp-Span-5200" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5201" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5202" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5203" class="MJXp-mo">[</span><span id="MJXp-Span-5204" class="MJXp-mi MJXp-italic">E</span><span id="MJXp-Span-5205" class="MJXp-mo">:</span><span id="MJXp-Span-5206" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-5207" class="MJXp-mo">]</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5208" class="MJXp-mo">(</span><span id="MJXp-Span-5209" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5210" class="MJXp-mrow"><span id="MJXp-Span-5211" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5212" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5213" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5214" class="MJXp-mi MJXp-italic">s</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5215" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5216" class="MJXp-mi MJXp-italic">o</span><span id="MJXp-Span-5217" class="MJXp-mrow"><span id="MJXp-Span-5218" class="MJXp-mo">/</span></span><span id="MJXp-Span-5219" class="MJXp-mo">[</span><span id="MJXp-Span-5220" class="MJXp-mi MJXp-italic">E</span><span id="MJXp-Span-5221" class="MJXp-mo">:</span><span id="MJXp-Span-5222" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-5223" class="MJXp-mo">]</span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5224" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-747-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-747" type="math/tex">M_{[E:H]}(\mathbb F_{s^{o/[E:H]}})</script></span> with respect to the basis <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5225" class="MJXp-math"><span id="MJXp-Span-5226" class="MJXp-mi MJXp-italic">B</span></span></span><span tabindex="0" id="MathJax-Element-748-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-748" type="math/tex">B</script></span> as stated in Corollary 29.8 in <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBR">[Rei03]</a>. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5227" class="MJXp-math"><span id="MJXp-Span-5228" class="MJXp-mi MJXp-italic">P</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5229" class="MJXp-mo">,</span><span id="MJXp-Span-5230" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5231" class="MJXp-mo">∈</span><span id="MJXp-Span-5232" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5233" class="MJXp-mi MJXp-italic">M</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5234" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5235" class="MJXp-mo">[</span><span id="MJXp-Span-5236" class="MJXp-mi MJXp-italic">E</span><span id="MJXp-Span-5237" class="MJXp-mo">:</span><span id="MJXp-Span-5238" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-5239" class="MJXp-mo">]</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5240" class="MJXp-mo">(</span><span id="MJXp-Span-5241" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5242" class="MJXp-mrow"><span id="MJXp-Span-5243" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5244" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5245" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5246" class="MJXp-mi MJXp-italic">s</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5247" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5248" class="MJXp-mi MJXp-italic">o</span><span id="MJXp-Span-5249" class="MJXp-mrow"><span id="MJXp-Span-5250" class="MJXp-mo">/</span></span><span id="MJXp-Span-5251" class="MJXp-mo">[</span><span id="MJXp-Span-5252" class="MJXp-mi MJXp-italic">E</span><span id="MJXp-Span-5253" class="MJXp-mo">:</span><span id="MJXp-Span-5254" class="MJXp-mi MJXp-italic">H</span><span id="MJXp-Span-5255" class="MJXp-mo">]</span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5256" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-749-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-749" type="math/tex">P,A\in M_{[E:H]}(\mathbb F_{s^{o/[E:H]}})</script></span> be the matrices</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5257" class="MJXp-math MJXp-display"><span id="MJXp-Span-5258" class="MJXp-mi MJXp-italic">P</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5259" class="MJXp-mo">=</span><span id="MJXp-Span-5260" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em; vertical-align: -2.306em;" id="MJXp-Span-5261" class="MJXp-mo"><span class="MJXp-right MJXp-scale2" style="font-size: 10.222em; margin-left: -0.25em;">(</span></span><span id="MJXp-Span-5262" class="MJXp-mtable"><span><span style="vertical-align: baseline;" id="MJXp-Span-5263" class="MJXp-mtr"><span style="text-align: right;" id="MJXp-Span-5264" class="MJXp-mtd"><span id="MJXp-Span-5265" class="MJXp-mn">1</span></span><span style="padding-left: 1em; text-align: right;" id="MJXp-Span-5266" class="MJXp-mtd"><span id="MJXp-Span-5267" class="MJXp-mn">1</span></span><span style="padding-left: 1em; text-align: right;" id="MJXp-Span-5268" class="MJXp-mtd"><span id="MJXp-Span-5269" class="MJXp-mn">1</span></span><span style="padding-left: 1em; text-align: right;" id="MJXp-Span-5270" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5271" class="MJXp-mo">⋯</span></span><span style="padding-left: 1em; text-align: right;" id="MJXp-Span-5272" class="MJXp-mtd"><span id="MJXp-Span-5273" class="MJXp-mn">1</span></span><span style="padding-left: 1em; text-align: right;" id="MJXp-Span-5274" class="MJXp-mtd"><span id="MJXp-Span-5275" class="MJXp-mn">1</span></span></span><span style="vertical-align: baseline;" id="MJXp-Span-5276" class="MJXp-mtr"><span style="padding-top: 0.4em; text-align: right;" id="MJXp-Span-5277" class="MJXp-mtd"><span id="MJXp-Span-5278" class="MJXp-mn">1</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5279" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0.111em;" id="MJXp-Span-5280" class="MJXp-mo">−</span><span id="MJXp-Span-5281" class="MJXp-mn">1</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5282" class="MJXp-mtd"><span id="MJXp-Span-5283" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5284" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5285" class="MJXp-mo">⋯</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5286" class="MJXp-mtd"><span id="MJXp-Span-5287" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5288" class="MJXp-mtd"><span id="MJXp-Span-5289" class="MJXp-mn">0</span></span></span><span style="vertical-align: baseline;" id="MJXp-Span-5290" class="MJXp-mtr"><span style="padding-top: 0.4em; text-align: right;" id="MJXp-Span-5291" class="MJXp-mtd"><span id="MJXp-Span-5292" class="MJXp-mn">1</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5293" class="MJXp-mtd"><span id="MJXp-Span-5294" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5295" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0.111em;" id="MJXp-Span-5296" class="MJXp-mo">−</span><span id="MJXp-Span-5297" class="MJXp-mn">1</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5298" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5299" class="MJXp-mo">⋯</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5300" class="MJXp-mtd"><span id="MJXp-Span-5301" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5302" class="MJXp-mtd"><span id="MJXp-Span-5303" class="MJXp-mn">0</span></span></span><span style="vertical-align: baseline;" id="MJXp-Span-5304" class="MJXp-mtr"><span style="padding-top: 0.4em; text-align: right;" id="MJXp-Span-5305" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5306" class="MJXp-mo">⋮</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5307" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5308" class="MJXp-mo">⋮</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5309" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5310" class="MJXp-mo">⋮</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5311" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5312" class="MJXp-mo">⋱</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5313" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5314" class="MJXp-mo">⋮</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5315" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5316" class="MJXp-mo">⋮</span></span></span><span style="vertical-align: baseline;" id="MJXp-Span-5317" class="MJXp-mtr"><span style="padding-top: 0.4em; text-align: right;" id="MJXp-Span-5318" class="MJXp-mtd"><span id="MJXp-Span-5319" class="MJXp-mn">1</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5320" class="MJXp-mtd"><span id="MJXp-Span-5321" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5322" class="MJXp-mtd"><span id="MJXp-Span-5323" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5324" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5325" class="MJXp-mo">⋯</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5326" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0.111em;" id="MJXp-Span-5327" class="MJXp-mo">−</span><span id="MJXp-Span-5328" class="MJXp-mn">1</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5329" class="MJXp-mtd"><span id="MJXp-Span-5330" class="MJXp-mn">0</span></span></span><span style="vertical-align: baseline;" id="MJXp-Span-5331" class="MJXp-mtr"><span style="padding-top: 0.4em; text-align: right;" id="MJXp-Span-5332" class="MJXp-mtd"><span id="MJXp-Span-5333" class="MJXp-mn">1</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5334" class="MJXp-mtd"><span id="MJXp-Span-5335" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5336" class="MJXp-mtd"><span id="MJXp-Span-5337" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5338" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5339" class="MJXp-mo">⋯</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5340" class="MJXp-mtd"><span id="MJXp-Span-5341" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: right;" id="MJXp-Span-5342" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0.111em;" id="MJXp-Span-5343" class="MJXp-mo">−</span><span id="MJXp-Span-5344" class="MJXp-mn">1</span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em; vertical-align: -2.306em;" id="MJXp-Span-5345" class="MJXp-mo"><span class="MJXp-right MJXp-scale2" style="font-size: 10.222em; margin-left: -0.25em;">)</span></span></span><span style="width: 1em; height: 0em;" id="MJXp-Span-5346" class="MJXp-mspace"></span><span id="MJXp-Span-5347" class="MJXp-mrow"><span id="MJXp-Span-5348" class="MJXp-mi">a</span><span id="MJXp-Span-5349" class="MJXp-mi">n</span><span id="MJXp-Span-5350" class="MJXp-mi">d</span></span><span style="width: 1em; height: 0em;" id="MJXp-Span-5351" class="MJXp-mspace"></span><span id="MJXp-Span-5352" class="MJXp-mi MJXp-italic">A</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5353" class="MJXp-mo">=</span><span id="MJXp-Span-5354" class="MJXp-mrow"><span style="margin-left: 0em; margin-right: 0em; vertical-align: -2.306em;" id="MJXp-Span-5355" class="MJXp-mo"><span class="MJXp-right MJXp-scale2" style="font-size: 10.222em; margin-left: -0.25em;">(</span></span><span id="MJXp-Span-5356" class="MJXp-mtable"><span><span style="vertical-align: baseline;" id="MJXp-Span-5357" class="MJXp-mtr"><span style="text-align: center;" id="MJXp-Span-5358" class="MJXp-mtd"><span id="MJXp-Span-5359" class="MJXp-mn">0</span></span><span style="padding-left: 1em; text-align: center;" id="MJXp-Span-5360" class="MJXp-mtd"><span id="MJXp-Span-5361" class="MJXp-mn">0</span></span><span style="padding-left: 1em; text-align: center;" id="MJXp-Span-5362" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5363" class="MJXp-mo">⋯</span></span><span style="padding-left: 1em; text-align: center;" id="MJXp-Span-5364" class="MJXp-mtd"><span id="MJXp-Span-5365" class="MJXp-mn">0</span></span><span style="padding-left: 1em; text-align: center;" id="MJXp-Span-5366" class="MJXp-mtd"><span id="MJXp-Span-5367" class="MJXp-mn">1</span></span></span><span style="vertical-align: baseline;" id="MJXp-Span-5368" class="MJXp-mtr"><span style="padding-top: 0.4em; text-align: center;" id="MJXp-Span-5369" class="MJXp-mtd"><span id="MJXp-Span-5370" class="MJXp-mn">1</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5371" class="MJXp-mtd"><span id="MJXp-Span-5372" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5373" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5374" class="MJXp-mo">⋯</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5375" class="MJXp-mtd"><span id="MJXp-Span-5376" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5377" class="MJXp-mtd"><span id="MJXp-Span-5378" class="MJXp-mn">0</span></span></span><span style="vertical-align: baseline;" id="MJXp-Span-5379" class="MJXp-mtr"><span style="padding-top: 0.4em; text-align: center;" id="MJXp-Span-5380" class="MJXp-mtd"><span id="MJXp-Span-5381" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5382" class="MJXp-mtd"><span id="MJXp-Span-5383" class="MJXp-mn">1</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5384" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5385" class="MJXp-mo">⋯</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5386" class="MJXp-mtd"><span id="MJXp-Span-5387" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5388" class="MJXp-mtd"><span id="MJXp-Span-5389" class="MJXp-mn">0</span></span></span><span style="vertical-align: baseline;" id="MJXp-Span-5390" class="MJXp-mtr"><span style="padding-top: 0.4em; text-align: center;" id="MJXp-Span-5391" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5392" class="MJXp-mo">⋮</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5393" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5394" class="MJXp-mo">⋮</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5395" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5396" class="MJXp-mo">⋱</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5397" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5398" class="MJXp-mo">⋮</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5399" class="MJXp-mtd"><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5400" class="MJXp-mo">⋮</span></span></span><span style="vertical-align: baseline;" id="MJXp-Span-5401" class="MJXp-mtr"><span style="padding-top: 0.4em; text-align: center;" id="MJXp-Span-5402" class="MJXp-mtd"><span id="MJXp-Span-5403" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5404" class="MJXp-mtd"><span id="MJXp-Span-5405" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5406" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5407" class="MJXp-mo">⋯</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5408" class="MJXp-mtd"><span id="MJXp-Span-5409" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5410" class="MJXp-mtd"><span id="MJXp-Span-5411" class="MJXp-mn">0</span></span></span><span style="vertical-align: baseline;" id="MJXp-Span-5412" class="MJXp-mtr"><span style="padding-top: 0.4em; text-align: center;" id="MJXp-Span-5413" class="MJXp-mtd"><span id="MJXp-Span-5414" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5415" class="MJXp-mtd"><span id="MJXp-Span-5416" class="MJXp-mn">0</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5417" class="MJXp-mtd"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5418" class="MJXp-mo">⋯</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5419" class="MJXp-mtd"><span id="MJXp-Span-5420" class="MJXp-mn">1</span></span><span style="padding-left: 1em; padding-top: 0.4em; text-align: center;" id="MJXp-Span-5421" class="MJXp-mtd"><span id="MJXp-Span-5422" class="MJXp-mn">0</span></span></span></span></span><span style="margin-left: 0em; margin-right: 0em; vertical-align: -2.306em;" id="MJXp-Span-5423" class="MJXp-mo"><span class="MJXp-right MJXp-scale2" style="font-size: 10.222em; margin-left: -0.25em;">)</span></span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5424" class="MJXp-mo">.</span></span></span><div class="MathJax_Display MathJax_Processing"><span tabindex="0" id="MathJax-Element-750-Frame" class="MathJax"></span></div><script id="MathJax-Element-750" type="math/tex; mode=display">
-P= \left( \begin{array}{rrrrrr}
-1 & 1 & 1 & \cdots & 1 & 1\\
-1 & -1 & 0 & \cdots & 0 & 0\\
-1 & 0 & -1 & \cdots & 0 & 0\\
-\vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\
-1 & 0 & 0 & \cdots & -1 & 0\\
-1 & 0 & 0 & \cdots & 0 & -1\\
-\end{array} \right)
-\quad {\rm and } \quad
-A= \left( \begin{array}{ccccc}
-0 & 0 & \cdots & 0 & 1\\
-1 & 0 & \cdots & 0 & 0\\
-0 & 1 & \cdots & 0 & 0\\
-\vdots & \vdots & \ddots & \vdots & \vdots\\
-0 & 0 & \cdots & 0 & 0\\
-0 & 0 & \cdots & 1 & 0\\
-\end{array} \right).
-</script></p>
-
-<p>Then</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5425" class="MJXp-math MJXp-display"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5426" class="MJXp-mo">{</span><span id="MJXp-Span-5427" class="MJXp-mi MJXp-italic">x</span><span id="MJXp-Span-5428" class="MJXp-mrow"><span id="MJXp-Span-5429" class="MJXp-munderover"><span><span class="MJXp-over"><span class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-5433" class="MJXp-mo">^</span></span><span class=""><span id="MJXp-Span-5430" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5431" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5432" class="MJXp-mn MJXp-script">1</span></span></span></span></span></span></span><span id="MJXp-Span-5434" class="MJXp-mi MJXp-italic">ε</span><span id="MJXp-Span-5435" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5436" class="MJXp-mi MJXp-italic">x</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5437" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5438" class="MJXp-mo">−</span><span id="MJXp-Span-5439" class="MJXp-mn">1</span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5440" class="MJXp-mo">∣</span><span id="MJXp-Span-5441" class="MJXp-mi MJXp-italic">x</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5442" class="MJXp-mo">∈</span><span id="MJXp-Span-5443" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5444" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5445" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5446" class="MJXp-mo">⟨</span><span id="MJXp-Span-5447" class="MJXp-mrow"><span id="MJXp-Span-5448" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5449" class="MJXp-mi MJXp-italic">x</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5450" class="MJXp-mi MJXp-italic MJXp-script">e</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5451" class="MJXp-mo">⟩</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5452" class="MJXp-mo">}</span></span></span><div class="MathJax_Display MathJax_Processing"><span tabindex="0" id="MathJax-Element-751-Frame" class="MathJax"></span></div><script id="MathJax-Element-751" type="math/tex; mode=display">
-\{x\widehat{T_1}\varepsilon x^{-1} \mid x\in T_2\langle{x_e}\rangle\}
-</script></p>
-
-<p>is a complete set of orthogonal primitive idempotents of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5453" class="MJXp-math"><span id="MJXp-Span-5454" class="MJXp-mrow"><span id="MJXp-Span-5455" class="MJXp-mi undefined">F</span></span><span id="MJXp-Span-5456" class="MJXp-mi MJXp-italic">G</span><span id="MJXp-Span-5457" class="MJXp-mi MJXp-italic">e</span></span></span><span tabindex="0" id="MathJax-Element-752-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-752" type="math/tex">\mathbb F G e</script></span> where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5458" class="MJXp-math"><span id="MJXp-Span-5459" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5460" class="MJXp-mi MJXp-italic">x</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5461" class="MJXp-mi MJXp-italic MJXp-script">e</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5462" class="MJXp-mo">=</span><span id="MJXp-Span-5463" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5464" class="MJXp-mi MJXp-italic">ψ</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5465" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5466" class="MJXp-mo">−</span><span id="MJXp-Span-5467" class="MJXp-mn">1</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5468" class="MJXp-mo">(</span><span id="MJXp-Span-5469" class="MJXp-mi MJXp-italic">P</span><span id="MJXp-Span-5470" class="MJXp-mi MJXp-italic">A</span><span id="MJXp-Span-5471" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5472" class="MJXp-mi MJXp-italic">P</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5473" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5474" class="MJXp-mo">−</span><span id="MJXp-Span-5475" class="MJXp-mn">1</span></span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5476" class="MJXp-mo">)</span></span></span><span tabindex="0" id="MathJax-Element-753-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-753" type="math/tex">x_e=\psi^{-1}(PAP^{-1})</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5477" class="MJXp-math"><span id="MJXp-Span-5478" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5479" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5480" class="MJXp-mn MJXp-script">1</span></span></span></span><span tabindex="0" id="MathJax-Element-754-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-754" type="math/tex">T_1</script></span> is a transversal of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5481" class="MJXp-math"><span id="MJXp-Span-5482" class="MJXp-mi MJXp-italic">H</span></span></span><span tabindex="0" id="MathJax-Element-755-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-755" type="math/tex">H</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5483" class="MJXp-math"><span id="MJXp-Span-5484" class="MJXp-mi MJXp-italic">E</span></span></span><span tabindex="0" id="MathJax-Element-756-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-756" type="math/tex">E</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5485" class="MJXp-math"><span id="MJXp-Span-5486" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5487" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5488" class="MJXp-mn MJXp-script">2</span></span></span></span><span tabindex="0" id="MathJax-Element-757-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-757" type="math/tex">T_2</script></span> is a right transversal of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5489" class="MJXp-math"><span id="MJXp-Span-5490" class="MJXp-mi MJXp-italic">E</span></span></span><span tabindex="0" id="MathJax-Element-758-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-758" type="math/tex">E</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5491" class="MJXp-math"><span id="MJXp-Span-5492" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-759-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-759" type="math/tex">G</script></span> (<a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html#biBOV2">[OGnt]</a>). By <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5493" class="MJXp-math"><span id="MJXp-Span-5494" class="MJXp-mrow"><span id="MJXp-Span-5495" class="MJXp-munderover"><span><span class="MJXp-over"><span class=""><span style="margin-left: 0px; margin-right: 0px;" id="MJXp-Span-5499" class="MJXp-mo">^</span></span><span class=""><span id="MJXp-Span-5496" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5497" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5498" class="MJXp-mn MJXp-script">1</span></span></span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-760-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-760" type="math/tex">\widehat{T_1}</script></span> we denote the element <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5500" class="MJXp-math"><span style="vertical-align: 0.25em;" id="MJXp-Span-5501" class="MJXp-mfrac"><span class="MJXp-box MJXp-script"><span id="MJXp-Span-5502" class="MJXp-mn">1</span></span><span style="margin-top: -0.9em;" class="MJXp-box"><span class="MJXp-denom"><span><span style="height: 1em; border-top: medium none; border-bottom: 1px solid; margin: 0.1em 0px;" class="MJXp-rule"></span></span><span><span class="MJXp-box MJXp-script"><span id="MJXp-Span-5503" class="MJXp-mrow"><span id="MJXp-Span-5504" class="MJXp-mo">|</span></span><span id="MJXp-Span-5505" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5506" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5507" class="MJXp-mn MJXp-script">1</span></span><span id="MJXp-Span-5508" class="MJXp-mrow"><span id="MJXp-Span-5509" class="MJXp-mo">|</span></span></span></span></span></span></span><span id="MJXp-Span-5510" class="MJXp-msubsup"><span style="margin-left: 0.111em; margin-right: 0.05em;" id="MJXp-Span-5511" class="MJXp-mo">∑</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5512" class="MJXp-mrow MJXp-script"><span id="MJXp-Span-5513" class="MJXp-mi MJXp-italic">t</span><span id="MJXp-Span-5514" class="MJXp-mo">∈</span><span id="MJXp-Span-5515" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5516" class="MJXp-mi MJXp-italic">T</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5517" class="MJXp-mn MJXp-script">1</span></span></span></span><span id="MJXp-Span-5518" class="MJXp-mrow"><span id="MJXp-Span-5519" class="MJXp-mi MJXp-italic">t</span></span></span></span><span tabindex="0" id="MathJax-Element-761-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-761" type="math/tex">\frac{1}{|T_1|}\sum_{t\in T_1}{t}</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5520" class="MJXp-math"><span id="MJXp-Span-5521" class="MJXp-mrow"><span id="MJXp-Span-5522" class="MJXp-mi undefined">F</span></span><span id="MJXp-Span-5523" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-762-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-762" type="math/tex">\mathbb F G</script></span>.</p>
-
-<p><a id="X856D7975810BF987" name="X856D7975810BF987"></a></p>
-
-<h4>9.21 <span class="Heading">Applications to coding theory</span></h4>
-
-<p>A <em>linear code</em> of length <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5524" class="MJXp-math"><span id="MJXp-Span-5525" class="MJXp-mi MJXp-italic">n</span></span></span><span tabindex="0" id="MathJax-Element-763-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-763" type="math/tex">n</script></span> and rank <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5526" class="MJXp-math"><span id="MJXp-Span-5527" class="MJXp-mi MJXp-italic">k</span></span></span><span tabindex="0" id="MathJax-Element-764-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-764" type="math/tex">k</script></span> is a linear subspace <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5528" class="MJXp-math"><span id="MJXp-Span-5529" class="MJXp-mi MJXp-italic">C</span></span></span><span tabindex="0" id="MathJax-Element-765-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-765" type="math/tex">C</script></span> with dimension <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5530" class="MJXp-math"><span id="MJXp-Span-5531" class="MJXp-mi MJXp-italic">k</span></span></span><span tabindex="0" id="MathJax-Element-766-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-766" type="math/tex">k</script></span> of the vector space <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5532" class="MJXp-math"><span id="MJXp-Span-5533" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5534" class="MJXp-mrow"><span id="MJXp-Span-5535" class="MJXp-mi undefined">F</span></span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-5537" class="MJXp-mi MJXp-italic">n</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-5536" class="MJXp-mi MJXp-italic">q</span></span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-767-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-767" type="math/tex">\mathbb F_q^n</script></span>. The standard basis of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5538" class="MJXp-math"><span id="MJXp-Span-5539" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5540" class="MJXp-mrow"><span id="MJXp-Span-5541" class="MJXp-mi undefined">F</span></span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-5543" class="MJXp-mi MJXp-italic">n</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-5542" class="MJXp-mi MJXp-italic">q</span></span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-768-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-768" type="math/tex">\mathbb F_q^n</script></span> is denoted by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5544" class="MJXp-math"><span id="MJXp-Span-5545" class="MJXp-mi MJXp-italic">E</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5546" class="MJXp-mo">=</span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5547" class="MJXp-mo">{</span><span id="MJXp-Span-5548" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5549" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5550" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5551" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5552" class="MJXp-mo">.</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5553" class="MJXp-mo">.</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5554" class="MJXp-mo">.</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5555" class="MJXp-mo">,</span><span id="MJXp-Span-5556" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5557" class="MJXp-mi MJXp-italic">e</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5558" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5559" class="MJXp-mo">}</span></span></span><span tabindex="0" id="MathJax-Element-769-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-769" type="math/tex">E=\{e_1,...,e_n\}</script></span>. The vectors in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5560" class="MJXp-math"><span id="MJXp-Span-5561" class="MJXp-mi MJXp-italic">C</span></span></span><span tabindex="0" id="MathJax-Element-770-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-770" type="math/tex">C</script></span> are called codewords, the size of a code is the number of codewords and equals <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5562" class="MJXp-math"><span id="MJXp-Span-5563" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5564" class="MJXp-mi MJXp-italic">q</span><span style="vertical-align: 0.5em;" id="MJXp-Span-5565" class="MJXp-mi MJXp-italic MJXp-script">k</span></span></span></span><span tabindex="0" id="MathJax-Element-771-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-771" type="math/tex">q^k</script></span>. The distance of a code is the minimum distance between distinct codewords, i.e. the number of elements in which they differ.</p>
-
-<p>For any group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5566" class="MJXp-math"><span id="MJXp-Span-5567" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-772-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-772" type="math/tex">G</script></span>, we denote by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5568" class="MJXp-math"><span id="MJXp-Span-5569" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5570" class="MJXp-mrow"><span id="MJXp-Span-5571" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5572" class="MJXp-mi MJXp-italic MJXp-script">q</span></span><span id="MJXp-Span-5573" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-773-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-773" type="math/tex">\mathbb F_q G</script></span> the group algebra over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5574" class="MJXp-math"><span id="MJXp-Span-5575" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-774-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-774" type="math/tex">G</script></span> with coefficients in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5576" class="MJXp-math"><span id="MJXp-Span-5577" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5578" class="MJXp-mrow"><span id="MJXp-Span-5579" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5580" class="MJXp-mi MJXp-italic MJXp-script">q</span></span></span></span><span tabindex="0" id="MathJax-Element-775-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-775" type="math/tex">\mathbb F_q</script></span>. If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5581" class="MJXp-math"><span id="MJXp-Span-5582" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-776-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-776" type="math/tex">G</script></span> is a group of order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5583" class="MJXp-math"><span id="MJXp-Span-5584" class="MJXp-mi MJXp-italic">n</span></span></span><span tabindex="0" id="MathJax-Element-777-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-777" type="math/tex">n</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5585" class="MJXp-math"><span id="MJXp-Span-5586" class="MJXp-mi MJXp-italic">C</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5587" class="MJXp-mo">⊆</span><span id="MJXp-Span-5588" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5589" class="MJXp-mrow"><span id="MJXp-Span-5590" class="MJXp-mi undefined">F</span></span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-5592" class="MJXp-mi MJXp-italic">n</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-5591" class="MJXp-mi MJXp-italic">q</span></span></span></span></span></span></span></span><span tabindex="0" id="MathJax-Element-778-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-778" type="math/tex">C\subseteq \mathbb F_q^n</script></span> is a linear code, then we say that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5593" class="MJXp-math"><span id="MJXp-Span-5594" class="MJXp-mi MJXp-italic">C</span></span></span><span tabindex="0" id="MathJax-Element-779-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-779" type="math/tex">C</script></span> is a left <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5595" class="MJXp-math"><span id="MJXp-Span-5596" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-780-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-780" type="math/tex">G</script></span>-code (respectively a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5597" class="MJXp-math"><span id="MJXp-Span-5598" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-781-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-781" type="math/tex">G</script></span>-code) if there is a bijection <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5599" class="MJXp-math"><span id="MJXp-Span-5600" class="MJXp-mi MJXp-italic">ϕ</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-5601" class="MJXp-mo">:</span><span id="MJXp-Span-5602" class="MJXp-mi MJXp-italic">E</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5603" class="MJXp-mo">→</span><span id="MJXp-Span-5604" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-782-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-782" type="math/tex">\phi:E\rightarrow G</script></span> such that the linear extension of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5605" class="MJXp-math"><span id="MJXp-Span-5606" class="MJXp-mi MJXp-italic">ϕ</span></span></span><span tabindex="0" id="MathJax-Element-783-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-783" type="math/tex">\phi</script></span> to an isomorphism <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5607" class="MJXp-math"><span id="MJXp-Span-5608" class="MJXp-mi MJXp-italic">ϕ</span><span style="margin-left: 0.111em; margin-right: 0.167em;" id="MJXp-Span-5609" class="MJXp-mo">:</span><span id="MJXp-Span-5610" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5611" class="MJXp-mrow"><span id="MJXp-Span-5612" class="MJXp-mi undefined">F</span></span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-5614" class="MJXp-mi MJXp-italic">n</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-5613" class="MJXp-mi MJXp-italic">q</span></span></span></span></span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5615" class="MJXp-mo">→</span><span id="MJXp-Span-5616" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5617" class="MJXp-mrow"><span id="MJXp-Span-5618" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5619" class="MJXp-mi MJXp-italic MJXp-script">q</span></span><span id="MJXp-Span-5620" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-784-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-784" type="math/tex">\phi:\mathbb F_q^n\rightarrow \mathbb F_q G</script></span> maps <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5621" class="MJXp-math"><span id="MJXp-Span-5622" class="MJXp-mi MJXp-italic">C</span></span></span><span tabindex="0" id="MathJax-Element-785-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-785" type="math/tex">C</script></span> to a left ideal (respectively a two-sided ideal) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5623" class="MJXp-math"><span id="MJXp-Span-5624" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5625" class="MJXp-mrow"><span id="MJXp-Span-5626" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5627" class="MJXp-mi MJXp-italic MJXp-script">q</span></span><span id="MJXp-Span-5628" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-786-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-786" type="math/tex">\mathbb F_q G</script></span>. A left <em>group code</em> (respectively a group code) is a linear code which is a left <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5629" class="MJXp-math"><span id="MJXp-Span-5630" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-787-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-787" type="math/tex">G</script></span>-code (respectively a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5631" class="MJXp-math"><span id="MJXp-Span-5632" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-788-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-788" type="math/tex">G</script></span>-code) for some group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5633" class="MJXp-math"><span id="MJXp-Span-5634" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-789-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-789" type="math/tex">G</script></span>.</p>
-
-<p>Since left ideals in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5635" class="MJXp-math"><span id="MJXp-Span-5636" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5637" class="MJXp-mrow"><span id="MJXp-Span-5638" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5639" class="MJXp-mi MJXp-italic MJXp-script">q</span></span><span id="MJXp-Span-5640" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-790-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-790" type="math/tex">\mathbb F_q G</script></span> are generated by idempotents, there is a one-one relation between (sums of) primitive idempotents of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5641" class="MJXp-math"><span id="MJXp-Span-5642" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5643" class="MJXp-mrow"><span id="MJXp-Span-5644" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5645" class="MJXp-mi MJXp-italic MJXp-script">q</span></span><span id="MJXp-Span-5646" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-791-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-791" type="math/tex">\mathbb F_q G</script></span> and left <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5647" class="MJXp-math"><span id="MJXp-Span-5648" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-792-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-792" type="math/tex">G</script></span>-codes over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5649" class="MJXp-math"><span id="MJXp-Span-5650" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5651" class="MJXp-mrow"><span id="MJXp-Span-5652" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5653" class="MJXp-mi MJXp-italic MJXp-script">q</span></span></span></span><span tabindex="0" id="MathJax-Element-793-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-793" type="math/tex">\mathbb F_q</script></span>.</p>
-
-<p>Note that each element <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5654" class="MJXp-math"><span id="MJXp-Span-5655" class="MJXp-mi MJXp-italic">c</span></span></span><span tabindex="0" id="MathJax-Element-794-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-794" type="math/tex">c</script></span> in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5656" class="MJXp-math"><span id="MJXp-Span-5657" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5658" class="MJXp-mrow"><span id="MJXp-Span-5659" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5660" class="MJXp-mi MJXp-italic MJXp-script">q</span></span><span id="MJXp-Span-5661" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-795-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-795" type="math/tex">\mathbb F_q G</script></span> is of the form <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5662" class="MJXp-math"><span id="MJXp-Span-5663" class="MJXp-mi MJXp-italic">c</span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5664" class="MJXp-mo">=</span><span id="MJXp-Span-5665" class="MJXp-msubsup"><span style="margin-left: 0.111em; margin-right: 0.05em;" id="MJXp-Span-5666" class="MJXp-mo">∑</span><span style="height: 1.86em; vertical-align: -0.64em;" class="MJXp-script-box"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span id="MJXp-Span-5671" class="MJXp-mi MJXp-italic">n</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span id="MJXp-Span-5667" class="MJXp-mrow"><span id="MJXp-Span-5668" class="MJXp-mi MJXp-italic">i</span><span id="MJXp-Span-5669" class="MJXp-mo">=</span><span id="MJXp-Span-5670" class="MJXp-mn">1</span></span></span></span></span></span></span><span id="MJXp-Span-5672" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5673" class="MJXp-mi MJXp-italic">f</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5674" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span id="MJXp-Span-5675" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5676" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5677" class="MJXp-mi MJXp-italic MJXp-script">i</span></span></span></span><span tabindex="0" id="MathJax-Element-796-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-796" type="math/tex">c=\sum_{i=1}^n f_i g_i</script></span>, where we fix an ordering <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5678" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5679" class="MJXp-mo">{</span><span id="MJXp-Span-5680" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5681" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5682" class="MJXp-mn MJXp-script">1</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5683" class="MJXp-mo">,</span><span id="MJXp-Span-5684" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5685" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5686" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5687" class="MJXp-mo">,</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5688" class="MJXp-mo">.</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5689" class="MJXp-mo">.</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5690" class="MJXp-mo">.</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5691" class="MJXp-mo">,</span><span id="MJXp-Span-5692" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5693" class="MJXp-mi MJXp-italic">g</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5694" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5695" class="MJXp-mo">}</span></span></span><span tabindex="0" id="MathJax-Element-797-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-797" type="math/tex">\{g_1,g_2,...,g_n \}</script></span> of the group elements of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5696" class="MJXp-math"><span id="MJXp-Span-5697" class="MJXp-mi MJXp-italic">G</span></span></span><span tabindex="0" id="MathJax-Element-798-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-798" type="math/tex">G</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5698" class="MJXp-math"><span id="MJXp-Span-5699" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5700" class="MJXp-mi MJXp-italic">f</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5701" class="MJXp-mi MJXp-italic MJXp-script">i</span></span><span style="margin-left: 0.333em; margin-right: 0.333em;" id="MJXp-Span-5702" class="MJXp-mo">∈</span><span id="MJXp-Span-5703" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5704" class="MJXp-mrow"><span id="MJXp-Span-5705" class="MJXp-mi undefined">F</span></span><span style="vertical-align: -0.4em;" id="MJXp-Span-5706" class="MJXp-mi MJXp-italic MJXp-script">q</span></span></span></span><span tabindex="0" id="MathJax-Element-799-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-799" type="math/tex">f_i\in \mathbb F_q</script></span>. If one looks at <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5707" class="MJXp-math"><span id="MJXp-Span-5708" class="MJXp-mi MJXp-italic">c</span></span></span><span tabindex="0" id="MathJax-Element-800-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-800" type="math/tex">c</script></span> as a codeword, one writes <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"><span id="MJXp-Span-5709" class="MJXp-math"><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5710" class="MJXp-mo">[</span><span id="MJXp-Span-5711" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5712" class="MJXp-mi MJXp-italic">f</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5713" class="MJXp-mn MJXp-script">1</span></span><span id="MJXp-Span-5714" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5715" class="MJXp-mi MJXp-italic">f</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5716" class="MJXp-mn MJXp-script">2</span></span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5717" class="MJXp-mo">.</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5718" class="MJXp-mo">.</span><span style="margin-left: 0em; margin-right: 0.222em;" id="MJXp-Span-5719" class="MJXp-mo">.</span><span id="MJXp-Span-5720" class="MJXp-msubsup"><span style="margin-right: 0.05em;" id="MJXp-Span-5721" class="MJXp-mi MJXp-italic">f</span><span style="vertical-align: -0.4em;" id="MJXp-Span-5722" class="MJXp-mi MJXp-italic MJXp-script">n</span></span><span style="margin-left: 0em; margin-right: 0em;" id="MJXp-Span-5723" class="MJXp-mo">]</span></span></span><span tabindex="0" id="MathJax-Element-801-Frame" class="MathJax MathJax_Processing"></span><script id="MathJax-Element-801" type="math/tex">[f_1 f_2 ... f_n]</script></span>.</p>
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href="http://www.gap-system.org/Manuals/doc/ref/chap64_mj.html">64</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap65_mj.html">65</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap66_mj.html">66</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap67_mj.html">67</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap68_mj.html">68</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap69_mj.html">69</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap70_mj.html">70</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html">71</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html">72</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html">73</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap74_mj.html">74</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap75_mj.html">75</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap76_mj.html">76</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap77_mj.html">77</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap78_mj.html">78</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap79_mj.html">79</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap80_mj.html">80</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap81_mj.html">81</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap82_mj.html">82</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap83_mj.html">83</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap84_mj.html">84</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap85_mj.html">85</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap86_mj.html">86</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap87_mj.html">87</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html">Bib</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chapInd_mj.html">Ind</a> </div>
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-<div class="ChapSects"><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7DFC03C187DE4841">18 <span class="Heading">Cyclotomic Numbers</span></a>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X79E25C3085AA568F">18.1 <span class="Heading">Operations for Cyclotomics</span></a>
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-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8631458886314588">18.1-1 E</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X863D1E017BC9EB7F">18.1-2 Cyclotomics</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X841C425281A6F775">18.1-3 IsCyclotomic</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X869750DA81EA0E67">18.1-4 IsIntegralCyclotomic</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7DD6B95F79321D23">18.1-5 Int</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7CBA6CB678E2B143">18.1-6 String</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X815D6EC57CBA9827">18.1-7 Conductor</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X81DD58BB81FB3426">18.1-8 AbsoluteValue</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7808ECF37AA9004D">18.1-9 RoundCyc</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7AE2933985BE4C3E">18.1-10 CoeffsCyc</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X803478CA7D2D830F">18.1-11 DenominatorCyc</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X785F2CAB805DE1BE">18.1-12 ExtRepOfObj</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7DDD51B983D5BC44">18.1-13 DescriptionOfRootOfUnity</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8712419182ECD8DD">18.1-14 IsGaussInt</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7E6CF4947D0A56F7">18.1-15 IsGaussRat</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7FE3D5637B5485D0">18.1-16 DefaultField</a></span>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7EE5FB7181125E02">18.2 <span class="Heading">Infinity and negative Infinity</span></a>
-</span>
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-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8511B8DF83324C27">18.2-1 IsInfinity</a></span>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7F66A62384329705">18.3 <span class="Heading">Comparisons of Cyclotomics</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7B242083873DD74F">18.4 <span class="Heading">ATLAS Irrationalities</span></a>
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-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X844D62597D9EFE3C">18.4-1 <span class="Heading">EB, EC, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2026;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-1-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-1" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.4em, 1000.9em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3" class="mo">…</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.269em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>…</mo></math></span></span><script id="MathJax-Element-1" type="math/tex">\ldots</script></span>, EH</span></a>
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-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X813CF4327C4B4D29">18.4-2 <span class="Heading">EI and ER</span></a>
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-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X842338F184E63894">18.4-3 <span class="Heading">EY, EX, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2026;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-2-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-4" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.4em, 1000.9em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-5" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-6" class="mo">…</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.269em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>…</mo></math></span></span><script id="MathJax-Element-2" type="math/tex">\ldots</script></span>, ES</span></a>
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-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X87CA3CA083D49AC7">18.4-4 <span class="Heading">EM, EL, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2026;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-3-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-7" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.4em, 1000.9em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-8" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-9" class="mo">…</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.269em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>…</mo></math></span></span><script id="MathJax-Element-3" type="math/tex">\ldots</script></span>, EJ</span></a>
-</span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X844F0EBF849EDEB3">18.4-5 NK</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X812E334E7A869D33">18.4-6 AtlasIrrationality</a></span>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X79FE34337DF2CD10">18.5 <span class="Heading">Galois Conjugacy of Cyclotomics</span></a>
-</span>
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-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X79EE9097783128C4">18.5-1 GaloisCyc</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7BE001A0811CD599">18.5-2 ComplexConjugate</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7E361C057E97CA66">18.5-3 StarCyc</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X84438F867B0CC299">18.5-4 Quadratic</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7DDDEC3F80543B7D">18.5-5 GaloisMat</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7BB9F5957AA8C082">18.5-6 RationalizedMat</a></span>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8557FC2D7ACD6105">18.6 <span class="Heading">Internally Represented Cyclotomics</span></a>
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-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7D3028777DE39709">18.6-1 SetCyclotomicsLimit</a></span>
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-<h3>18 <span class="Heading">Cyclotomic Numbers</span></h3>
-
-<p><strong class="pkg">GAP</strong> admits computations in abelian extension fields of the rational number field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-4-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-10" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.72em, 2.853em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-11" class="mrow"><span id="MathJax-Span-12" class="texatom"><span id="MathJax-Span-13" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-14" class="mo">â„š</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.147em; border-left: 0px solid; width: 0px; height: 1.088em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow></math></span></span><script id="MathJax-Element-4" type="math/tex">â„š</script></span>, that is fields with abelian Galois group over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-5-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-15" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.72em, 2.853em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-16" class="mrow"><span id="MathJax-Span-17" class="texatom"><span id="MathJax-Span-18" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-19" class="mo">â„š</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.147em; border-left: 0px solid; width: 0px; height: 1.088em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow></math></span></span><script id="MathJax-Element-5" type="math/tex">â„š</script></span>. These fields are subfields of <em>cyclotomic fields</em> <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mo stretchy="false">(</mo><msub><mi>e</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-6-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.016em; display: inline-block;" role="math" id="MathJax-Span-20" class="math"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.26em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-21" class="mrow"><span id="MathJax-Span-22" class="texatom"><span id="MathJax-Span-23" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-24" class="mo">â„š</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-25" class="mo">(</span><span id="MathJax-Span-26" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-27" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-28" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-29" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mo stretchy="false">(</mo><msub><mi>e</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-6" type="math/tex">â„š(e_n)</script></span> where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mi>n</mi></msub><mo>=</mo><mi>exp</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mn>2</mn><mi>&#x03C0;</mi><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-7-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.203em; display: inline-block;" role="math" id="MathJax-Span-30" class="math"><span style="display: inline-block; position: relative; width: 6.298em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1006.25em, 3.005em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-31" class="mrow"><span id="MathJax-Span-32" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-33" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-34" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-35" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-36" class="mi">exp</span><span id="MathJax-Span-37" class="mo"></span><span style="font-family: STIXGeneral;" id="MathJax-Span-38" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-39" class="mn">2</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-40" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-41" class="mi">i</span><span id="MathJax-Span-42" class="texatom"><span id="MathJax-Span-43" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-44" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-45" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-46" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.345em; border-left: 0px solid; width: 0px; height: 1.286em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mi>n</mi></msub><mo>=</mo><mi>exp</mi><mo>â¡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>Ï€</mi><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-7" type="math/tex">e_n = \exp(2 \pi i/n)</script></span> is a primitive complex <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-8-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-47" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-48" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-49" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-8" type="math/tex">n</script></span>-th root of unity. The elements of these fields are called <em>cyclotomics</em>.</p>
-
-<p>Information concerning operations for domains of cyclotomics, for
-example certain integral bases of fields of cyclotomics, can be found in
- Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80510B5880521FDC"><span class="RefLink">60</span></a>. For more general operations that take a field extension as a –possibly optional– argument, e.g., <code class="func">Trace</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html#X7DD17EB581200AD6"><span class="RefLink">58.3-5</span></a>) or <code class="func">Coefficients</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap61_mj.html#X80B32F667BF6AFD8"><span class="RefLink">61.6-3</span></a>), see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html#X80A8E676814A19FD"><span class="RefLink">58</span></a>.</p>
-
-<p><a id="X79E25C3085AA568F" name="X79E25C3085AA568F"></a></p>
-
-<h4>18.1 <span class="Heading">Operations for Cyclotomics</span></h4>
-
-<p><a id="X8631458886314588" name="X8631458886314588"></a></p>
-
-<h5>18.1-1 E</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ E</code>( <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p><code class="func">E</code> returns the primitive <var class="Arg">n</var>-th root of unity <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mi>n</mi></msub><mo>=</mo><mi>exp</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mn>2</mn><mi>&#x03C0;</mi><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-9-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.203em; display: inline-block;" role="math" id="MathJax-Span-50" class="math"><span style="display: inline-block; position: relative; width: 6.298em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1006.25em, 3.005em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-51" class="mrow"><span id="MathJax-Span-52" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-53" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-54" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-55" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-56" class="mi">exp</span><span id="MathJax-Span-57" class="mo"></span><span style="font-family: STIXGeneral;" id="MathJax-Span-58" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-59" class="mn">2</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-60" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-61" class="mi">i</span><span id="MathJax-Span-62" class="texatom"><span id="MathJax-Span-63" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-64" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-65" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-66" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.345em; border-left: 0px solid; width: 0px; height: 1.286em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mi>n</mi></msub><mo>=</mo><mi>exp</mi><mo>â¡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>Ï€</mi><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-9" type="math/tex">e_n = \exp(2\pi i/n)</script></span>.
- Cyclotomics are usually entered as sums of roots of unity, with
-rational coefficients, and irrational cyclotomics are displayed in such a
- way. (For special cyclotomics, see <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7B242083873DD74F"><span class="RefLink">18.4</span></a>.)</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">E(9); E(9)^3; E(6); E(12) / 3;</span>
--E(9)^4-E(9)^7
-E(3)
--E(3)^2
--1/3*E(12)^7
-</pre></div>
-
-<p>A particular basis is used to express cyclotomics, see <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>; note that <code class="code">E(9)</code> is <em>not</em> a basis element, as the above example shows.</p>
-
-<p><a id="X863D1E017BC9EB7F" name="X863D1E017BC9EB7F"></a></p>
-
-<h5>18.1-2 Cyclotomics</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Cyclotomics</code></td><td class="tdright">( global variable )</td></tr></tbody></table></div>
-<p>is the domain of all cyclotomics.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">E(9) in Cyclotomics; 37 in Cyclotomics; true in Cyclotomics;</span>
-true
-true
-false
-</pre></div>
-
-<p>As the cyclotomics are field elements, the usual arithmetic operators <code class="code">+</code>, <code class="code">-</code>, <code class="code">*</code> and <code class="code">/</code> (and <code class="code">^</code> to take powers by integers) are applicable. Note that <code class="code">^</code> does <em>not</em> denote the conjugation of group elements, so it is <em>not</em>
- possible to explicitly construct groups of cyclotomics. (However, it is
- possible to compute the inverse and the multiplicative order of a
-nonzero cyclotomic.) Also, taking the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-10-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-67" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.43em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-68" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-69" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script id="MathJax-Element-10" type="math/tex">k</script></span>-th power of a root of unity <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-11-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.516em; display: inline-block;" role="math" id="MathJax-Span-70" class="math"><span style="display: inline-block; position: relative; width: 0.385em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.38em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-71" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-72" class="mi">z</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 0.787em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></span></span><script id="MathJax-Element-11" type="math/tex">z</script></span> defines a Galois automorphism if and only if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-12-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-73" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.43em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-74" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-75" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script id="MathJax-Element-12" type="math/tex">k</script></span> is coprime to the conductor (see <code class="func">Conductor</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>)) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-13-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.516em; display: inline-block;" role="math" id="MathJax-Span-76" class="math"><span style="display: inline-block; position: relative; width: 0.385em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.38em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-77" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-78" class="mi">z</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 0.787em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></span></span><script id="MathJax-Element-13" type="math/tex">z</script></span>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">E(5) + E(3); (E(5) + E(5)^4) ^ 2; E(5) / E(3); E(5) * E(3);</span>
--E(15)^2-2*E(15)^8-E(15)^11-E(15)^13-E(15)^14
--2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
-E(15)^13
-E(15)^8
-<span class="GAPprompt">gap></span> <span class="GAPinput">Order( E(5) ); Order( 1+E(5) );</span>
-5
-infinity
-</pre></div>
-
-<p><a id="X841C425281A6F775" name="X841C425281A6F775"></a></p>
-
-<h5>18.1-3 IsCyclotomic</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsCyclotomic</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsCyc</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<p>Every object in the family <code class="code">CyclotomicsFamily</code> lies in the category <code class="func">IsCyclotomic</code>. This covers integers, rationals, proper cyclotomics, the object <code class="func">infinity</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>), and unknowns (see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap74_mj.html#X7C1FAB6280A02CCB"><span class="RefLink">74</span></a>). All these objects except <code class="func">infinity</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) and unknowns lie also in the category <code class="func">IsCyc</code>, <code class="func">infinity</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) lies in (and can be detected from) the category <code class="func">IsInfinity</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>), and unknowns lie in <code class="func">IsUnknown</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap74_mj.html#X828556067E069B6D"><span class="RefLink">74.1-3</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomic(0); IsCyclotomic(1/2*E(3)); IsCyclotomic( infinity );</span>
-true
-true
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyc(0); IsCyc(1/2*E(3)); IsCyc( infinity );</span>
-true
-true
-false
-</pre></div>
-
-<p><a id="X869750DA81EA0E67" name="X869750DA81EA0E67"></a></p>
-
-<h5>18.1-4 IsIntegralCyclotomic</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsIntegralCyclotomic</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<p>A cyclotomic is called <em>integral</em> or a <em>cyclotomic integer</em>
- if all coefficients of its minimal polynomial over the rationals are
-integers. Since the underlying basis of the external representation of
-cyclotomics is an integral basis (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>),
- the subring of cyclotomic integers in a cyclotomic field is formed by
-those cyclotomics for which the external representation is a list of
-integers. For example, square roots of integers are cyclotomic integers
-(see <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7B242083873DD74F"><span class="RefLink">18.4</span></a>),
- any root of unity is a cyclotomic integer, character values are always
-cyclotomic integers, but all rationals which are not integers are not
-cyclotomic integers.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">r:= ER( 5 ); # The square root of 5 ...</span>
-E(5)-E(5)^2-E(5)^3+E(5)^4
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsIntegralCyclotomic( r ); # ... is a cyclotomic integer.</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">r2:= 1/2 * r; # This is not a cyclotomic integer, ...</span>
-1/2*E(5)-1/2*E(5)^2-1/2*E(5)^3+1/2*E(5)^4
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsIntegralCyclotomic( r2 );</span>
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">r3:= 1/2 * r - 1/2; # ... but this is one.</span>
-E(5)+E(5)^4
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsIntegralCyclotomic( r3 );</span>
-true
-</pre></div>
-
-<p><a id="X7DD6B95F79321D23" name="X7DD6B95F79321D23"></a></p>
-
-<h5>18.1-5 Int</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Int</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( method )</td></tr></tbody></table></div>
-<p>The operation <code class="func">Int</code> can be used to find a cyclotomic integer near to an arbitrary cyclotomic, by applying <code class="func">Int</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap14_mj.html#X87CA734380B5F68C"><span class="RefLink">14.2-3</span></a>) to the coefficients.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">Int( E(5)+1/2*E(5)^2 ); Int( 2/3*E(7)-3/2*E(4) );</span>
-E(5)
--E(4)
-</pre></div>
-
-<p><a id="X7CBA6CB678E2B143" name="X7CBA6CB678E2B143"></a></p>
-
-<h5>18.1-6 String</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ String</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( method )</td></tr></tbody></table></div>
-<p>The operation <code class="func">String</code> returns for a cyclotomic <var class="Arg">cyc</var> a string corresponding to the way the cyclotomic is printed by <code class="func">ViewObj</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) and <code class="func">PrintObj</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">String( E(5)+1/2*E(5)^2 ); String( 17/3 );</span>
-"E(5)+1/2*E(5)^2"
-"17/3"
-</pre></div>
-
-<p><a id="X815D6EC57CBA9827" name="X815D6EC57CBA9827"></a></p>
-
-<h5>18.1-7 Conductor</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Conductor</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Conductor</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For an element <var class="Arg">cyc</var> of a cyclotomic field, <code class="func">Conductor</code> returns the smallest integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-14-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-79" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-80" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-81" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-14" type="math/tex">n</script></span> such that <var class="Arg">cyc</var> is contained in the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-15-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-82" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-83" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-84" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-15" type="math/tex">n</script></span>-th cyclotomic field. For a collection <var class="Arg">C</var> of cyclotomics (for example a dense list of cyclotomics or a field of cyclotomics), <code class="func">Conductor</code> returns the smallest integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-16-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-85" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-86" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-87" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-16" type="math/tex">n</script></span> such that all elements of <var class="Arg">C</var> are contained in the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-17-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-88" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-89" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-90" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-17" type="math/tex">n</script></span>-th cyclotomic field.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">Conductor( 0 ); Conductor( E(10) ); Conductor( E(12) );</span>
-1
-5
-12
-</pre></div>
-
-<p><a id="X81DD58BB81FB3426" name="X81DD58BB81FB3426"></a></p>
-
-<h5>18.1-8 AbsoluteValue</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ AbsoluteValue</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>returns the absolute value of a cyclotomic number <var class="Arg">cyc</var>. At the moment only methods for rational numbers exist.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">AbsoluteValue(-3);</span>
-3
-</pre></div>
-
-<p><a id="X7808ECF37AA9004D" name="X7808ECF37AA9004D"></a></p>
-
-<h5>18.1-9 RoundCyc</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ RoundCyc</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>is a cyclotomic integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-18-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.516em; display: inline-block;" role="math" id="MathJax-Span-91" class="math"><span style="display: inline-block; position: relative; width: 0.385em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.38em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-92" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-93" class="mi">z</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 0.787em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></span></span><script id="MathJax-Element-18" type="math/tex">z</script></span> (see <code class="func">IsIntegralCyclotomic</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X869750DA81EA0E67"><span class="RefLink">18.1-4</span></a>)) near to the cyclotomic <var class="Arg">cyc</var> in the following sense: Let <code class="code">c</code> be the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-19-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-94" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-95" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-96" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-19" type="math/tex">i</script></span>-th coefficient in the external representation (see <code class="func">CoeffsCyc</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7AE2933985BE4C3E"><span class="RefLink">18.1-10</span></a>)) of <var class="Arg">cyc</var>. Then the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-20-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-97" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-98" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-99" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-20" type="math/tex">i</script></span>-th coefficient in the external representation of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-21-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.516em; display: inline-block;" role="math" id="MathJax-Span-100" class="math"><span style="display: inline-block; position: relative; width: 0.385em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.38em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-101" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-102" class="mi">z</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 0.787em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></span></span><script id="MathJax-Element-21" type="math/tex">z</script></span> is <code class="code">Int( c + 1/2 )</code> or <code class="code">Int( c - 1/2 )</code>, depending on whether <code class="code">c</code> is nonnegative or negative, respectively.</p>
-
-<p>Expressed in terms of the Zumbroich basis (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>), rounding the coefficients of <var class="Arg">cyc</var> w.r.t. this basis to the nearest integer yields the coefficients of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-22-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.516em; display: inline-block;" role="math" id="MathJax-Span-103" class="math"><span style="display: inline-block; position: relative; width: 0.385em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.38em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-104" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-105" class="mi">z</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 0.787em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></span></span><script id="MathJax-Element-22" type="math/tex">z</script></span>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">RoundCyc( E(5)+1/2*E(5)^2 ); RoundCyc( 2/3*E(7)+3/2*E(4) );</span>
-E(5)+E(5)^2
--2*E(28)^3+E(28)^4-2*E(28)^11-2*E(28)^15-2*E(28)^19-2*E(28)^23
- -2*E(28)^27
-</pre></div>
-
-<p><a id="X7AE2933985BE4C3E" name="X7AE2933985BE4C3E"></a></p>
-
-<h5>18.1-10 CoeffsCyc</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CoeffsCyc</code>( <var class="Arg">cyc</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">cyc</var> be a cyclotomic with conductor <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-23-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-106" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-107" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-108" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-23" type="math/tex">n</script></span> (see <code class="func">Conductor</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>)). If <var class="Arg">N</var> is not a multiple of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-24-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-109" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-110" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-111" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-24" type="math/tex">n</script></span> then <code class="func">CoeffsCyc</code> returns <code class="keyw">fail</code> because <var class="Arg">cyc</var> cannot be expressed in terms of <var class="Arg">N</var>-th roots of unity. Otherwise <code class="func">CoeffsCyc</code> returns a list of length <var class="Arg">N</var> with entry at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-25-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-112" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.848em, 1000.29em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-113" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-114" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.242em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></span></span><script id="MathJax-Element-25" type="math/tex">j</script></span> equal to the coefficient of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>exp</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mn>2</mn><mi>&#x03C0;</mi><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-26-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 9.016em; display: inline-block;" role="math" id="MathJax-Span-115" class="math"><span style="display: inline-block; position: relative; width: 6.923em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1006.88em, 3.005em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-116" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-117" class="mi">exp</span><span id="MathJax-Span-118" class="mo"></span><span style="font-family: STIXGeneral;" id="MathJax-Span-119" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-120" class="mn">2</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-121" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-122" class="mi">i</span><span style="font-family: STIXGeneral;" id="MathJax-Span-123" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-124" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-125" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-126" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-127" class="mo">)</span><span id="MathJax-Span-128" class="texatom"><span id="MathJax-Span-129" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-130" class="mo">/</span></span></span><span id="MathJax-Span-131" class="texatom"><span id="MathJax-Span-132" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-133" class="mtext">N</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-134" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.345em; border-left: 0px solid; width: 0px; height: 1.286em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>exp</mi><mo>â¡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>Ï€</mi><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-26" type="math/tex">\exp(2 \pi i (j-1)/\textit{N})</script></span> if this root belongs to the <var class="Arg">N</var>-th Zumbroich basis (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>), and equal to zero otherwise. So we have <var class="Arg">cyc</var> = <code class="code">CoeffsCyc(</code> <var class="Arg">cyc</var>, <var class="Arg">N</var> <code class="code">) * List( [1..</code><var class="Arg">N</var><code class="code">], j -> E(</code><var class="Arg">N</var><code class="code">)^(j-1) )</code>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">cyc:= E(5)+E(5)^2;</span>
-E(5)+E(5)^2
-<span class="GAPprompt">gap></span> <span class="GAPinput">CoeffsCyc( cyc, 5 ); CoeffsCyc( cyc, 15 ); CoeffsCyc( cyc, 7 );</span>
-[ 0, 1, 1, 0, 0 ]
-[ 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0 ]
-fail
-</pre></div>
-
-<p><a id="X803478CA7D2D830F" name="X803478CA7D2D830F"></a></p>
-
-<h5>18.1-11 DenominatorCyc</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ DenominatorCyc</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>For a cyclotomic number <var class="Arg">cyc</var> (see <code class="func">IsCyclotomic</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X841C425281A6F775"><span class="RefLink">18.1-3</span></a>)), this function returns the smallest positive integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-27-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-135" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-136" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-137" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-27" type="math/tex">n</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-28-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-138" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-139" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-140" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-28" type="math/tex">n</script></span><code class="code"> * </code><var class="Arg">cyc</var> is a cyclotomic integer (see <code class="func">IsIntegralCyclotomic</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X869750DA81EA0E67"><span class="RefLink">18.1-4</span></a>)). For rational numbers <var class="Arg">cyc</var>, the result is the same as that of <code class="func">DenominatorRat</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap17_mj.html#X81F6B5877A81E727"><span class="RefLink">17.2-5</span></a>).</p>
-
-<p><a id="X785F2CAB805DE1BE" name="X785F2CAB805DE1BE"></a></p>
-
-<h5>18.1-12 ExtRepOfObj</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ExtRepOfObj</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( method )</td></tr></tbody></table></div>
-<p>The external representation of a cyclotomic <var class="Arg">cyc</var> with conductor <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-29-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-141" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-142" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-143" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-29" type="math/tex">n</script></span> (see <code class="func">Conductor</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>) is the list returned by <code class="func">CoeffsCyc</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7AE2933985BE4C3E"><span class="RefLink">18.1-10</span></a>), called with <var class="Arg">cyc</var> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-30-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-144" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-145" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-146" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-30" type="math/tex">n</script></span>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">ExtRepOfObj( E(5) ); CoeffsCyc( E(5), 5 );</span>
-[ 0, 1, 0, 0, 0 ]
-[ 0, 1, 0, 0, 0 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">CoeffsCyc( E(5), 15 );</span>
-[ 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0 ]
-</pre></div>
-
-<p><a id="X7DDD51B983D5BC44" name="X7DDD51B983D5BC44"></a></p>
-
-<h5>18.1-13 DescriptionOfRootOfUnity</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ DescriptionOfRootOfUnity</code>( <var class="Arg">root</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Given a cyclotomic <var class="Arg">root</var> that is known to be a root of unity (this is <em>not</em> checked), <code class="func">DescriptionOfRootOfUnity</code> returns a list <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>n</mi><mo>,</mo><mi>e</mi><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-31-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.641em; display: inline-block;" role="math" id="MathJax-Span-147" class="math"><span style="display: inline-block; position: relative; width: 2.019em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.838em, 1001.93em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-148" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-149" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-150" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-151" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-152" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-153" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.188em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>n</mi><mo>,</mo><mi>e</mi><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-31" type="math/tex">[ n, e ]</script></span> of coprime positive integers such that <var class="Arg">root</var> <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-32-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-154" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.114em, 1000.63em, 2.768em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-155" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-156" class="mo">=</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.036em; border-left: 0px solid; width: 0px; height: 0.601em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math></span></span><script id="MathJax-Element-32" type="math/tex">=</script></span> <code class="code">E</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo><mi>e</mi></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-33-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.016em; display: inline-block;" role="math" id="MathJax-Span-157" class="math"><span style="display: inline-block; position: relative; width: 1.538em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1001.54em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-158" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-159" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-160" class="mi">n</span><span id="MathJax-Span-161" class="msubsup"><span style="display: inline-block; position: relative; width: 0.722em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-162" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-163" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo><mi>e</mi></msup></math></span></span><script id="MathJax-Element-33" type="math/tex">(n)^e</script></span> holds.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">E(9); DescriptionOfRootOfUnity( E(9) );</span>
--E(9)^4-E(9)^7
-[ 9, 1 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">DescriptionOfRootOfUnity( -E(3) );</span>
-[ 6, 5 ]
-</pre></div>
-
-<p><a id="X8712419182ECD8DD" name="X8712419182ECD8DD"></a></p>
-
-<h5>18.1-14 IsGaussInt</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsGaussInt</code>( <var class="Arg">x</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p><code class="func">IsGaussInt</code> returns <code class="keyw">true</code> if the object <var class="Arg">x</var> is a Gaussian integer (see <code class="func">GaussianIntegers</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80BD5EAB879F096E"><span class="RefLink">60.5-1</span></a>)), and <code class="keyw">false</code> otherwise. Gaussian integers are of the form <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-34-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.766em; display: inline-block;" role="math" id="MathJax-Span-164" class="math"><span style="display: inline-block; position: relative; width: 2.115em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1002.09em, 2.829em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-165" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-166" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-167" class="mo">+</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-168" class="mi">b</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.116em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi></math></span></span><script id="MathJax-Element-34" type="math/tex">a + b</script></span><code class="code">*E(4)</code>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-35-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-169" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-170" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-171" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.713em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script id="MathJax-Element-35" type="math/tex">a</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-36-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-172" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.45em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-173" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-174" class="mi">b</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script id="MathJax-Element-36" type="math/tex">b</script></span> are integers.</p>
-
-<p><a id="X7E6CF4947D0A56F7" name="X7E6CF4947D0A56F7"></a></p>
-
-<h5>18.1-15 IsGaussRat</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsGaussRat</code>( <var class="Arg">x</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p><code class="func">IsGaussRat</code> returns <code class="keyw">true</code> if the object <var class="Arg">x</var> is a Gaussian rational (see <code class="func">GaussianRationals</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X82F53C65802FF551"><span class="RefLink">60.1-3</span></a>)), and <code class="keyw">false</code> otherwise. Gaussian rationals are of the form <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-37-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.766em; display: inline-block;" role="math" id="MathJax-Span-175" class="math"><span style="display: inline-block; position: relative; width: 2.115em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1002.09em, 2.829em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-176" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-177" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-178" class="mo">+</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-179" class="mi">b</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.116em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi></math></span></span><script id="MathJax-Element-37" type="math/tex">a + b</script></span><code class="code">*E(4)</code>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-38-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-180" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-181" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-182" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.713em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script id="MathJax-Element-38" type="math/tex">a</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-39-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-183" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.45em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-184" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-185" class="mi">b</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script id="MathJax-Element-39" type="math/tex">b</script></span> are rationals.</p>
-
-<p><a id="X7FE3D5637B5485D0" name="X7FE3D5637B5485D0"></a></p>
-
-<h5>18.1-16 DefaultField</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ DefaultField</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p><code class="func">DefaultField</code> for cyclotomics is defined to return the smallest <em>cyclotomic</em> field containing the given elements.</p>
-
-<p>Note that <code class="func">Field</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html#X871AA7D58263E9AC"><span class="RefLink">58.1-3</span></a>)
- returns the smallest field containing all given elements, which need
-not be a cyclotomic field. In both cases, the fields represent vector
-spaces over the rationals (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">Field( E(5)+E(5)^4 ); DefaultField( E(5)+E(5)^4 );</span>
-NF(5,[ 1, 4 ])
-CF(5)
-</pre></div>
-
-<p><a id="X7EE5FB7181125E02" name="X7EE5FB7181125E02"></a></p>
-
-<h4>18.2 <span class="Heading">Infinity and negative Infinity</span></h4>
-
-<p><a id="X8511B8DF83324C27" name="X8511B8DF83324C27"></a></p>
-
-<h5>18.2-1 IsInfinity</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsInfinity</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsNegInfinity</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ infinity</code></td><td class="tdright">( global variable )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ -infinity</code></td><td class="tdright">( global variable )</td></tr></tbody></table></div>
-<p><code class="func">infinity</code> and <code class="func">-infinity</code> are special <strong class="pkg">GAP</strong> objects that lie in <code class="code">CyclotomicsFamily</code>. They are larger or smaller than all other objects in this family respectively. <code class="func">infinity</code> is mainly used as return value of operations such as <code class="func">Size</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap30_mj.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>) and <code class="func">Dimension</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap57_mj.html#X7E6926C6850E7C4E"><span class="RefLink">57.3-3</span></a>) for infinite and infinite dimensional domains, respectively.</p>
-
-<p>Some arithmetic operations are provided for convenience when using <code class="func">infinity</code> and <code class="func">-infinity</code> as top and bottom element respectively.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">-infinity + 1;</span>
--infinity
-<span class="GAPprompt">gap></span> <span class="GAPinput">infinity + infinity;</span>
-infinity
-</pre></div>
-
-<p>Often it is useful to distinguish <code class="func">infinity</code> from "proper" cyclotomics. For that, <code class="func">infinity</code> lies in the category <code class="func">IsInfinity</code> but not in <code class="func">IsCyc</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X841C425281A6F775"><span class="RefLink">18.1-3</span></a>), and the other cyclotomics lie in the category <code class="func">IsCyc</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X841C425281A6F775"><span class="RefLink">18.1-3</span></a>) but not in <code class="func">IsInfinity</code>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">s:= Size( Rationals );</span>
-infinity
-<span class="GAPprompt">gap></span> <span class="GAPinput">s = infinity; IsCyclotomic( s ); IsCyc( s ); IsInfinity( s );</span>
-true
-true
-false
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">s in Rationals; s > 17;</span>
-false
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">Set( [ s, 2, s, E(17), s, 19 ] );</span>
-[ 2, 19, E(17), infinity ]
-</pre></div>
-
-<p><a id="X7F66A62384329705" name="X7F66A62384329705"></a></p>
-
-<h4>18.3 <span class="Heading">Comparisons of Cyclotomics</span></h4>
-
-<p>To compare cyclotomics, the operators <code class="code"><</code>, <code class="code"><=</code>, <code class="code">=</code>, <code class="code">>=</code>, <code class="code">></code>, and <code class="code"><></code> can be used, the result will be <code class="keyw">true</code> if the first operand is smaller, smaller or equal, equal, larger or equal, larger, or unequal, respectively, and <code class="keyw">false</code> otherwise.</p>
-
-<p>Cyclotomics are ordered as follows: The relation between rationals is
- the natural one, rationals are smaller than irrational cyclotomics, and
- <code class="func">infinity</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) is the largest cyclotomic. For two irrational cyclotomics with different conductors (see <code class="func">Conductor</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>)),
- the one with smaller conductor is regarded as smaller. Two irrational
-cyclotomics with same conductor are compared via their external
-representation (see <code class="func">ExtRepOfObj</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X785F2CAB805DE1BE"><span class="RefLink">18.1-12</span></a>)).</p>
-
-<p>For comparisons of cyclotomics and other <strong class="pkg">GAP</strong> objects, see Section <a href="http://www.gap-system.org/Manuals/doc/ref/chap4_mj.html#X7A274A1F8553B7E6"><span class="RefLink">4.12</span></a>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">E(5) < E(6); # the latter value has conductor 3</span>
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">E(3) < E(3)^2; # both have conductor 3, compare the ext. repr.</span>
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">3 < E(3); E(5) < E(7);</span>
-true
-true
-</pre></div>
-
-<p><a id="X7B242083873DD74F" name="X7B242083873DD74F"></a></p>
-
-<h4>18.4 <span class="Heading">ATLAS Irrationalities</span></h4>
-
-<p><a id="X844D62597D9EFE3C" name="X844D62597D9EFE3C"></a></p>
-
-<h5>18.4-1 <span class="Heading">EB, EC, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2026;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-40-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-186" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.4em, 1000.9em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-187" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-188" class="mo">…</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.269em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>…</mo></math></span></span><script id="MathJax-Element-40" type="math/tex">\ldots</script></span>, EH</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EB</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EC</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ED</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EE</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EF</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EG</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EH</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>For a positive integer <var class="Arg">N</var>, let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-41-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.828em; display: inline-block;" role="math" id="MathJax-Span-189" class="math"><span style="display: inline-block; position: relative; width: 1.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1001.35em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-190" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-191" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-192" class="mo">=</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 0.787em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo></math></span></span><script id="MathJax-Element-41" type="math/tex">z =</script></span> <code class="code">E(</code><var class="Arg">N</var><code class="code">)</code> <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>exp</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mn>2</mn><mi>&#x03C0;</mi><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-42-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 6.891em; display: inline-block;" role="math" id="MathJax-Span-193" class="math"><span style="display: inline-block; position: relative; width: 5.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1005.24em, 3.005em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-194" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-195" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-196" class="mi">exp</span><span id="MathJax-Span-197" class="mo"></span><span style="font-family: STIXGeneral;" id="MathJax-Span-198" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-199" class="mn">2</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-200" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-201" class="mi">i</span><span id="MathJax-Span-202" class="texatom"><span id="MathJax-Span-203" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-204" class="mo">/</span></span></span><span id="MathJax-Span-205" class="texatom"><span id="MathJax-Span-206" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-207" class="mtext">N</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-208" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.345em; border-left: 0px solid; width: 0px; height: 1.286em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>exp</mi><mo>â¡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>Ï€</mi><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-42" type="math/tex">= \exp(2 \pi i/\textit{N})</script></span>. The following so-called <em>atomic irrationalities</em> (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCCN85">[CCNPW85, Chapter 7, Section 10]</a>) can be entered using functions. (Note that the values are not necessary irrational.)</p>
-
-<div class="pcenter"><table class="GAPDocTablenoborder">
-<tbody><tr>
-<td class="tdleft"><code class="code">EB(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-43-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-209" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1001.06em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-210" class="mrow"><span id="MathJax-Span-211" class="msubsup"><span style="display: inline-block; position: relative; width: 1.047em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-212" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-213" class="texatom"><span id="MathJax-Span-214" class="mrow"><span id="MathJax-Span-215" class="texatom"><span id="MathJax-Span-216" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-217" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.222em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-43" type="math/tex">b_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>2</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-44-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.203em; display: inline-block;" role="math" id="MathJax-Span-218" class="math"><span style="display: inline-block; position: relative; width: 5.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.108em, 1005.5em, 3.808em, -1000em); top: -3.221em; left: 0em;"><span id="MathJax-Span-219" class="mrow"><span id="MathJax-Span-220" class="mrow"><span style="vertical-align: -0.246em;" id="MathJax-Span-221" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">(</span></span></span><span id="MathJax-Span-222" class="munderover"><span style="display: inline-block; position: relative; width: 2.299em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-223" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.39em, 4.145em, -1000em); top: -4.478em; left: 0.914em;"><span id="MathJax-Span-224" class="texatom"><span id="MathJax-Span-225" class="mrow"><span id="MathJax-Span-226" class="texatom"><span id="MathJax-Span-227" class="mrow"><span id="MathJax-Span-228" class="texatom"><span id="MathJax-Span-229" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-230" class="mtext">N</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-231" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-232" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.11em, 4.281em, -1000em); top: -3.694em; left: 0.914em;"><span id="MathJax-Span-233" class="texatom"><span id="MathJax-Span-234" class="mrow"><span id="MathJax-Span-235" class="texatom"><span id="MathJax-Span-236" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-237" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-238" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-239" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-240" class="msubsup"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-241" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-242" class="texatom"><span id="MathJax-Span-243" class="mrow"><span id="MathJax-Span-244" class="texatom"><span id="MathJax-Span-245" class="mrow"><span id="MathJax-Span-246" class="msubsup"><span style="display: inline-block; position: relative; width: 0.551em; height: 0px;"><span style="position: absolute; clip: rect(3.385em, 1000.2em, 4.281em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-247" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.258em; left: 0.247em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-248" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.246em;" id="MathJax-Span-249" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">)</span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-250" class="texatom"><span id="MathJax-Span-251" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-252" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-253" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 3.221em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.638em; border-left: 0px solid; width: 0px; height: 1.96em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>−</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>2</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script id="MathJax-Element-44" type="math/tex">\left( \sum_{{j = 1}}^{{\textit{N}-1}} z^{{j^2}} \right) / 2</script></span></td>
-<td class="tdcenter">,</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2261;</mo><mn>1</mn><mspace width="0.444em" /><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em" /><mn>2</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-45-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.328em; display: inline-block;" role="math" id="MathJax-Span-254" class="math"><span style="display: inline-block; position: relative; width: 6.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1006.35em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-255" class="mrow"><span id="MathJax-Span-256" class="texatom"><span id="MathJax-Span-257" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-258" class="mtext">N</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-259" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-260" class="mn">1</span><span id="MathJax-Span-261" class="TeXmathchoice"><span style="height: 0em; vertical-align: 0em; width: 0.5em; display: inline-block; overflow: hidden;" id="MathJax-Span-262" class="mspace"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-263" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-264" class="mi">mod</span><span style="height: 0em; vertical-align: 0em; width: 0.375em; display: inline-block; overflow: hidden;" id="MathJax-Span-265" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-266" class="mn">2</span><span style="font-family: STIXGeneral;" id="MathJax-Span-267" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>2</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-45" type="math/tex">\textit{N} \equiv 1 \pmod{2}</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EC(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-46-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-268" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.01em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-269" class="mrow"><span id="MathJax-Span-270" class="msubsup"><span style="display: inline-block; position: relative; width: 0.991em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-271" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-272" class="texatom"><span id="MathJax-Span-273" class="mrow"><span id="MathJax-Span-274" class="texatom"><span id="MathJax-Span-275" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-276" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 0.907em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-46" type="math/tex">c_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>3</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-47-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.203em; display: inline-block;" role="math" id="MathJax-Span-277" class="math"><span style="display: inline-block; position: relative; width: 5.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.108em, 1005.46em, 3.808em, -1000em); top: -3.221em; left: 0em;"><span id="MathJax-Span-278" class="mrow"><span id="MathJax-Span-279" class="mrow"><span style="vertical-align: -0.246em;" id="MathJax-Span-280" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">(</span></span></span><span id="MathJax-Span-281" class="munderover"><span style="display: inline-block; position: relative; width: 2.299em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-282" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.39em, 4.145em, -1000em); top: -4.478em; left: 0.914em;"><span id="MathJax-Span-283" class="texatom"><span id="MathJax-Span-284" class="mrow"><span id="MathJax-Span-285" class="texatom"><span id="MathJax-Span-286" class="mrow"><span id="MathJax-Span-287" class="texatom"><span id="MathJax-Span-288" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-289" class="mtext">N</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-290" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-291" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.11em, 4.281em, -1000em); top: -3.694em; left: 0.914em;"><span id="MathJax-Span-292" class="texatom"><span id="MathJax-Span-293" class="mrow"><span id="MathJax-Span-294" class="texatom"><span id="MathJax-Span-295" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-296" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-297" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-298" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-299" class="msubsup"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-300" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-301" class="texatom"><span id="MathJax-Span-302" class="mrow"><span id="MathJax-Span-303" class="texatom"><span id="MathJax-Span-304" class="mrow"><span id="MathJax-Span-305" class="msubsup"><span style="display: inline-block; position: relative; width: 0.551em; height: 0px;"><span style="position: absolute; clip: rect(3.385em, 1000.2em, 4.281em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-306" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.258em; left: 0.247em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-307" class="mn">3</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.246em;" id="MathJax-Span-308" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">)</span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-309" class="texatom"><span id="MathJax-Span-310" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-311" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-312" class="mn">3</span></span><span style="display: inline-block; width: 0px; height: 3.221em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.638em; border-left: 0px solid; width: 0px; height: 1.96em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>−</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>3</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn></math></span></span><script id="MathJax-Element-47" type="math/tex">\left( \sum_{{j = 1}}^{{\textit{N}-1}} z^{{j^3}} \right) / 3</script></span></td>
-<td class="tdcenter">,</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2261;</mo><mn>1</mn><mspace width="0.444em" /><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em" /><mn>3</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-48-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.328em; display: inline-block;" role="math" id="MathJax-Span-313" class="math"><span style="display: inline-block; position: relative; width: 6.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1006.35em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-314" class="mrow"><span id="MathJax-Span-315" class="texatom"><span id="MathJax-Span-316" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-317" class="mtext">N</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-318" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-319" class="mn">1</span><span id="MathJax-Span-320" class="TeXmathchoice"><span style="height: 0em; vertical-align: 0em; width: 0.5em; display: inline-block; overflow: hidden;" id="MathJax-Span-321" class="mspace"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-322" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-323" class="mi">mod</span><span style="height: 0em; vertical-align: 0em; width: 0.375em; display: inline-block; overflow: hidden;" id="MathJax-Span-324" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-325" class="mn">3</span><span style="font-family: STIXGeneral;" id="MathJax-Span-326" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>3</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-48" type="math/tex">\textit{N} \equiv 1 \pmod{3}</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">ED(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-49-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-327" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1001.06em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-328" class="mrow"><span id="MathJax-Span-329" class="msubsup"><span style="display: inline-block; position: relative; width: 1.047em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.53em, 4.148em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-330" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-331" class="texatom"><span id="MathJax-Span-332" class="mrow"><span id="MathJax-Span-333" class="texatom"><span id="MathJax-Span-334" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-335" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.222em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-49" type="math/tex">d_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>4</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>4</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-50-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.203em; display: inline-block;" role="math" id="MathJax-Span-336" class="math"><span style="display: inline-block; position: relative; width: 5.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.108em, 1005.5em, 3.808em, -1000em); top: -3.221em; left: 0em;"><span id="MathJax-Span-337" class="mrow"><span id="MathJax-Span-338" class="mrow"><span style="vertical-align: -0.246em;" id="MathJax-Span-339" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">(</span></span></span><span id="MathJax-Span-340" class="munderover"><span style="display: inline-block; position: relative; width: 2.299em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-341" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.39em, 4.145em, -1000em); top: -4.478em; left: 0.914em;"><span id="MathJax-Span-342" class="texatom"><span id="MathJax-Span-343" class="mrow"><span id="MathJax-Span-344" class="texatom"><span id="MathJax-Span-345" class="mrow"><span id="MathJax-Span-346" class="texatom"><span id="MathJax-Span-347" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-348" class="mtext">N</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-349" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-350" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.11em, 4.281em, -1000em); top: -3.694em; left: 0.914em;"><span id="MathJax-Span-351" class="texatom"><span id="MathJax-Span-352" class="mrow"><span id="MathJax-Span-353" class="texatom"><span id="MathJax-Span-354" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-355" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-356" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-357" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-358" class="msubsup"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-359" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-360" class="texatom"><span id="MathJax-Span-361" class="mrow"><span id="MathJax-Span-362" class="texatom"><span id="MathJax-Span-363" class="mrow"><span id="MathJax-Span-364" class="msubsup"><span style="display: inline-block; position: relative; width: 0.551em; height: 0px;"><span style="position: absolute; clip: rect(3.385em, 1000.2em, 4.281em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-365" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.258em; left: 0.247em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-366" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.246em;" id="MathJax-Span-367" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">)</span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-368" class="texatom"><span id="MathJax-Span-369" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-370" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-371" class="mn">4</span></span><span style="display: inline-block; width: 0px; height: 3.221em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.638em; border-left: 0px solid; width: 0px; height: 1.96em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>−</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>4</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>4</mn></math></span></span><script id="MathJax-Element-50" type="math/tex">\left( \sum_{{j = 1}}^{{\textit{N}-1}} z^{{j^4}} \right) / 4</script></span></td>
-<td class="tdcenter">,</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2261;</mo><mn>1</mn><mspace width="0.444em" /><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em" /><mn>4</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-51-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.328em; display: inline-block;" role="math" id="MathJax-Span-372" class="math"><span style="display: inline-block; position: relative; width: 6.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1006.35em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-373" class="mrow"><span id="MathJax-Span-374" class="texatom"><span id="MathJax-Span-375" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-376" class="mtext">N</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-377" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-378" class="mn">1</span><span id="MathJax-Span-379" class="TeXmathchoice"><span style="height: 0em; vertical-align: 0em; width: 0.5em; display: inline-block; overflow: hidden;" id="MathJax-Span-380" class="mspace"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-381" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-382" class="mi">mod</span><span style="height: 0em; vertical-align: 0em; width: 0.375em; display: inline-block; overflow: hidden;" id="MathJax-Span-383" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-384" class="mn">4</span><span style="font-family: STIXGeneral;" id="MathJax-Span-385" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>4</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-51" type="math/tex">\textit{N} \equiv 1 \pmod{4}</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EE(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-52-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-386" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.01em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-387" class="mrow"><span id="MathJax-Span-388" class="msubsup"><span style="display: inline-block; position: relative; width: 0.991em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-389" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-390" class="texatom"><span id="MathJax-Span-391" class="mrow"><span id="MathJax-Span-392" class="texatom"><span id="MathJax-Span-393" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-394" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 0.907em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-52" type="math/tex">e_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>5</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>5</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-53-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.203em; display: inline-block;" role="math" id="MathJax-Span-395" class="math"><span style="display: inline-block; position: relative; width: 5.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.102em, 1005.47em, 3.808em, -1000em); top: -3.221em; left: 0em;"><span id="MathJax-Span-396" class="mrow"><span id="MathJax-Span-397" class="mrow"><span style="vertical-align: -0.246em;" id="MathJax-Span-398" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">(</span></span></span><span id="MathJax-Span-399" class="munderover"><span style="display: inline-block; position: relative; width: 2.299em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-400" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.39em, 4.145em, -1000em); top: -4.478em; left: 0.914em;"><span id="MathJax-Span-401" class="texatom"><span id="MathJax-Span-402" class="mrow"><span id="MathJax-Span-403" class="texatom"><span id="MathJax-Span-404" class="mrow"><span id="MathJax-Span-405" class="texatom"><span id="MathJax-Span-406" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-407" class="mtext">N</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-408" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-409" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.11em, 4.281em, -1000em); top: -3.694em; left: 0.914em;"><span id="MathJax-Span-410" class="texatom"><span id="MathJax-Span-411" class="mrow"><span id="MathJax-Span-412" class="texatom"><span id="MathJax-Span-413" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-414" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-415" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-416" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-417" class="msubsup"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-418" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-419" class="texatom"><span id="MathJax-Span-420" class="mrow"><span id="MathJax-Span-421" class="texatom"><span id="MathJax-Span-422" class="mrow"><span id="MathJax-Span-423" class="msubsup"><span style="display: inline-block; position: relative; width: 0.551em; height: 0px;"><span style="position: absolute; clip: rect(3.385em, 1000.2em, 4.281em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-424" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.258em; left: 0.247em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-425" class="mn">5</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.246em;" id="MathJax-Span-426" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">)</span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-427" class="texatom"><span id="MathJax-Span-428" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-429" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-430" class="mn">5</span></span><span style="display: inline-block; width: 0px; height: 3.221em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.638em; border-left: 0px solid; width: 0px; height: 1.968em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>−</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>5</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>5</mn></math></span></span><script id="MathJax-Element-53" type="math/tex">\left( \sum_{{j = 1}}^{{\textit{N}-1}} z^{{j^5}} \right) / 5</script></span></td>
-<td class="tdcenter">,</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2261;</mo><mn>1</mn><mspace width="0.444em" /><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em" /><mn>5</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-54-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.328em; display: inline-block;" role="math" id="MathJax-Span-431" class="math"><span style="display: inline-block; position: relative; width: 6.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.812em, 1006.35em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-432" class="mrow"><span id="MathJax-Span-433" class="texatom"><span id="MathJax-Span-434" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-435" class="mtext">N</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-436" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-437" class="mn">1</span><span id="MathJax-Span-438" class="TeXmathchoice"><span style="height: 0em; vertical-align: 0em; width: 0.5em; display: inline-block; overflow: hidden;" id="MathJax-Span-439" class="mspace"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-440" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-441" class="mi">mod</span><span style="height: 0em; vertical-align: 0em; width: 0.375em; display: inline-block; overflow: hidden;" id="MathJax-Span-442" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-443" class="mn">5</span><span style="font-family: STIXGeneral;" id="MathJax-Span-444" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.249em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>5</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-54" type="math/tex">\textit{N} \equiv 1 \pmod{5}</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EF(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-55-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.078em; display: inline-block;" role="math" id="MathJax-Span-445" class="math"><span style="display: inline-block; position: relative; width: 0.817em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.63em, 1000.82em, 2.803em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-446" class="mrow"><span id="MathJax-Span-447" class="msubsup"><span style="display: inline-block; position: relative; width: 0.825em; height: 0px;"><span style="position: absolute; clip: rect(3.168em, 1000.42em, 4.342em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic; text-rendering: optimizelegibility;" id="MathJax-Span-448" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span id="MathJax-Span-449" class="texatom"><span id="MathJax-Span-450" class="mrow"><span id="MathJax-Span-451" class="texatom"><span id="MathJax-Span-452" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-453" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.276em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-55" type="math/tex">f_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>6</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>6</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-56-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.203em; display: inline-block;" role="math" id="MathJax-Span-454" class="math"><span style="display: inline-block; position: relative; width: 5.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.104em, 1005.5em, 3.808em, -1000em); top: -3.221em; left: 0em;"><span id="MathJax-Span-455" class="mrow"><span id="MathJax-Span-456" class="mrow"><span style="vertical-align: -0.246em;" id="MathJax-Span-457" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">(</span></span></span><span id="MathJax-Span-458" class="munderover"><span style="display: inline-block; position: relative; width: 2.299em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-459" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.39em, 4.145em, -1000em); top: -4.478em; left: 0.914em;"><span id="MathJax-Span-460" class="texatom"><span id="MathJax-Span-461" class="mrow"><span id="MathJax-Span-462" class="texatom"><span id="MathJax-Span-463" class="mrow"><span id="MathJax-Span-464" class="texatom"><span id="MathJax-Span-465" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-466" class="mtext">N</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-467" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-468" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.11em, 4.281em, -1000em); top: -3.694em; left: 0.914em;"><span id="MathJax-Span-469" class="texatom"><span id="MathJax-Span-470" class="mrow"><span id="MathJax-Span-471" class="texatom"><span id="MathJax-Span-472" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-473" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-474" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-475" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-476" class="msubsup"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-477" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-478" class="texatom"><span id="MathJax-Span-479" class="mrow"><span id="MathJax-Span-480" class="texatom"><span id="MathJax-Span-481" class="mrow"><span id="MathJax-Span-482" class="msubsup"><span style="display: inline-block; position: relative; width: 0.551em; height: 0px;"><span style="position: absolute; clip: rect(3.385em, 1000.2em, 4.281em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-483" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.258em; left: 0.247em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-484" class="mn">6</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.246em;" id="MathJax-Span-485" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">)</span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-486" class="texatom"><span id="MathJax-Span-487" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-488" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-489" class="mn">6</span></span><span style="display: inline-block; width: 0px; height: 3.221em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.638em; border-left: 0px solid; width: 0px; height: 1.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>−</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>6</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>6</mn></math></span></span><script id="MathJax-Element-56" type="math/tex">\left( \sum_{{j = 1}}^{{\textit{N}-1}} z^{{j^6}} \right) / 6</script></span></td>
-<td class="tdcenter">,</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2261;</mo><mn>1</mn><mspace width="0.444em" /><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em" /><mn>6</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-57-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.328em; display: inline-block;" role="math" id="MathJax-Span-490" class="math"><span style="display: inline-block; position: relative; width: 6.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.816em, 1006.35em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-491" class="mrow"><span id="MathJax-Span-492" class="texatom"><span id="MathJax-Span-493" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-494" class="mtext">N</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-495" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-496" class="mn">1</span><span id="MathJax-Span-497" class="TeXmathchoice"><span style="height: 0em; vertical-align: 0em; width: 0.5em; display: inline-block; overflow: hidden;" id="MathJax-Span-498" class="mspace"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-499" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-500" class="mi">mod</span><span style="height: 0em; vertical-align: 0em; width: 0.375em; display: inline-block; overflow: hidden;" id="MathJax-Span-501" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-502" class="mn">6</span><span style="font-family: STIXGeneral;" id="MathJax-Span-503" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.244em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>6</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-57" type="math/tex">\textit{N} \equiv 1 \pmod{6}</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EG(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-58-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-504" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.06em, 2.802em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-505" class="mrow"><span id="MathJax-Span-506" class="msubsup"><span style="display: inline-block; position: relative; width: 1.047em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-507" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-508" class="texatom"><span id="MathJax-Span-509" class="mrow"><span id="MathJax-Span-510" class="texatom"><span id="MathJax-Span-511" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-512" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 0.966em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-58" type="math/tex">g_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>7</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>7</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-59-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.203em; display: inline-block;" role="math" id="MathJax-Span-513" class="math"><span style="display: inline-block; position: relative; width: 5.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.111em, 1005.48em, 3.808em, -1000em); top: -3.221em; left: 0em;"><span id="MathJax-Span-514" class="mrow"><span id="MathJax-Span-515" class="mrow"><span style="vertical-align: -0.246em;" id="MathJax-Span-516" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">(</span></span></span><span id="MathJax-Span-517" class="munderover"><span style="display: inline-block; position: relative; width: 2.299em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-518" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.39em, 4.145em, -1000em); top: -4.478em; left: 0.914em;"><span id="MathJax-Span-519" class="texatom"><span id="MathJax-Span-520" class="mrow"><span id="MathJax-Span-521" class="texatom"><span id="MathJax-Span-522" class="mrow"><span id="MathJax-Span-523" class="texatom"><span id="MathJax-Span-524" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-525" class="mtext">N</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-526" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-527" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.11em, 4.281em, -1000em); top: -3.694em; left: 0.914em;"><span id="MathJax-Span-528" class="texatom"><span id="MathJax-Span-529" class="mrow"><span id="MathJax-Span-530" class="texatom"><span id="MathJax-Span-531" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-532" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-533" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-534" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-535" class="msubsup"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-536" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-537" class="texatom"><span id="MathJax-Span-538" class="mrow"><span id="MathJax-Span-539" class="texatom"><span id="MathJax-Span-540" class="mrow"><span id="MathJax-Span-541" class="msubsup"><span style="display: inline-block; position: relative; width: 0.551em; height: 0px;"><span style="position: absolute; clip: rect(3.385em, 1000.2em, 4.281em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-542" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.258em; left: 0.247em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-543" class="mn">7</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.246em;" id="MathJax-Span-544" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">)</span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-545" class="texatom"><span id="MathJax-Span-546" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-547" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-548" class="mn">7</span></span><span style="display: inline-block; width: 0px; height: 3.221em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.638em; border-left: 0px solid; width: 0px; height: 1.956em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>−</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>7</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>7</mn></math></span></span><script id="MathJax-Element-59" type="math/tex">\left( \sum_{{j = 1}}^{{\textit{N}-1}} z^{{j^7}} \right) / 7</script></span></td>
-<td class="tdcenter">,</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2261;</mo><mn>1</mn><mspace width="0.444em" /><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em" /><mn>7</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-60-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.328em; display: inline-block;" role="math" id="MathJax-Span-549" class="math"><span style="display: inline-block; position: relative; width: 6.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1006.35em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-550" class="mrow"><span id="MathJax-Span-551" class="texatom"><span id="MathJax-Span-552" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-553" class="mtext">N</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-554" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-555" class="mn">1</span><span id="MathJax-Span-556" class="TeXmathchoice"><span style="height: 0em; vertical-align: 0em; width: 0.5em; display: inline-block; overflow: hidden;" id="MathJax-Span-557" class="mspace"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-558" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-559" class="mi">mod</span><span style="height: 0em; vertical-align: 0em; width: 0.375em; display: inline-block; overflow: hidden;" id="MathJax-Span-560" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-561" class="mn">7</span><span style="font-family: STIXGeneral;" id="MathJax-Span-562" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>7</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-60" type="math/tex">\textit{N} \equiv 1 \pmod{7}</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EH(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-61-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-563" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1001.06em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-564" class="mrow"><span id="MathJax-Span-565" class="msubsup"><span style="display: inline-block; position: relative; width: 1.047em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.48em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-566" class="mi">h</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-567" class="texatom"><span id="MathJax-Span-568" class="mrow"><span id="MathJax-Span-569" class="texatom"><span id="MathJax-Span-570" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-571" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.222em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-61" type="math/tex">h_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>8</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>8</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-62-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.203em; display: inline-block;" role="math" id="MathJax-Span-572" class="math"><span style="display: inline-block; position: relative; width: 5.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.108em, 1005.47em, 3.808em, -1000em); top: -3.221em; left: 0em;"><span id="MathJax-Span-573" class="mrow"><span id="MathJax-Span-574" class="mrow"><span style="vertical-align: -0.246em;" id="MathJax-Span-575" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">(</span></span></span><span id="MathJax-Span-576" class="munderover"><span style="display: inline-block; position: relative; width: 2.299em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-577" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.39em, 4.145em, -1000em); top: -4.478em; left: 0.914em;"><span id="MathJax-Span-578" class="texatom"><span id="MathJax-Span-579" class="mrow"><span id="MathJax-Span-580" class="texatom"><span id="MathJax-Span-581" class="mrow"><span id="MathJax-Span-582" class="texatom"><span id="MathJax-Span-583" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-584" class="mtext">N</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-585" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-586" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.11em, 4.281em, -1000em); top: -3.694em; left: 0.914em;"><span id="MathJax-Span-587" class="texatom"><span id="MathJax-Span-588" class="mrow"><span id="MathJax-Span-589" class="texatom"><span id="MathJax-Span-590" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-591" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-592" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-593" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-594" class="msubsup"><span style="display: inline-block; position: relative; width: 1.015em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-595" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-596" class="texatom"><span id="MathJax-Span-597" class="mrow"><span id="MathJax-Span-598" class="texatom"><span id="MathJax-Span-599" class="mrow"><span id="MathJax-Span-600" class="msubsup"><span style="display: inline-block; position: relative; width: 0.551em; height: 0px;"><span style="position: absolute; clip: rect(3.385em, 1000.2em, 4.281em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-601" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.258em; left: 0.247em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-602" class="mn">8</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.246em;" id="MathJax-Span-603" class="mo"><span><span style="font-size: 110%; font-family: STIXSizeOneSym;">)</span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-604" class="texatom"><span id="MathJax-Span-605" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-606" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-607" class="mn">8</span></span><span style="display: inline-block; width: 0px; height: 3.221em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.638em; border-left: 0px solid; width: 0px; height: 1.96em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>−</mo><mn>1</mn></mrow></mrow></munderover><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>j</mi><mn>8</mn></msup></mrow></mrow></msup><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>8</mn></math></span></span><script id="MathJax-Element-62" type="math/tex">\left( \sum_{{j = 1}}^{{\textit{N}-1}} z^{{j^8}} \right) / 8</script></span></td>
-<td class="tdcenter">,</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2261;</mo><mn>1</mn><mspace width="0.444em" /><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em" /><mn>8</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-63-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.328em; display: inline-block;" role="math" id="MathJax-Span-608" class="math"><span style="display: inline-block; position: relative; width: 6.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1006.35em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-609" class="mrow"><span id="MathJax-Span-610" class="texatom"><span id="MathJax-Span-611" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-612" class="mtext">N</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-613" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-614" class="mn">1</span><span id="MathJax-Span-615" class="TeXmathchoice"><span style="height: 0em; vertical-align: 0em; width: 0.5em; display: inline-block; overflow: hidden;" id="MathJax-Span-616" class="mspace"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-617" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-618" class="mi">mod</span><span style="height: 0em; vertical-align: 0em; width: 0.375em; display: inline-block; overflow: hidden;" id="MathJax-Span-619" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-620" class="mn">8</span><span style="font-family: STIXGeneral;" id="MathJax-Span-621" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>8</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-63" type="math/tex">\textit{N} \equiv 1 \pmod{8}</script></span></td>
-</tr>
-</tbody></table><br><p> </p><br>
-</div>
-
-<p>(Note that in <code class="code">EC(</code><var class="Arg">N</var><code class="code">)</code>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2026;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-64-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-622" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.4em, 1000.9em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-623" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-624" class="mo">…</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.269em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>…</mo></math></span></span><script id="MathJax-Element-64" type="math/tex">\ldots</script></span>, <code class="code">EH(</code><var class="Arg">N</var><code class="code">)</code>, <var class="Arg">N</var> must be a prime.)</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">EB(5); EB(9);</span>
-E(5)+E(5)^4
-1
-</pre></div>
-
-<p><a id="X813CF4327C4B4D29" name="X813CF4327C4B4D29"></a></p>
-
-<h5>18.4-2 <span class="Heading">EI and ER</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EI</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ER</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>For a rational number <var class="Arg">N</var>, <code class="func">ER</code> returns the square root <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msqrt></math>" style="position: relative;" tabindex="0" id="MathJax-Element-65-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.891em; display: inline-block;" role="math" id="MathJax-Span-625" class="math"><span style="display: inline-block; position: relative; width: 1.442em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.497em, 1001.48em, 2.739em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-626" class="mrow"><span id="MathJax-Span-627" class="msqrt"><span style="display: inline-block; position: relative; width: 1.429em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-628" class="mrow"><span id="MathJax-Span-629" class="texatom"><span id="MathJax-Span-630" class="mrow"><span id="MathJax-Span-631" class="texatom"><span id="MathJax-Span-632" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-633" class="mtext">N</span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1000.69em, 3.365em, -1000em); top: -3.981em; left: 0.737em;"><span style="display: inline-block; position: relative; width: 0.692em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0.192em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.858em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.248em; border-left: 0px solid; width: 0px; height: 1.365em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msqrt></math></span></span><script id="MathJax-Element-65" type="math/tex">\sqrt{{\textit{N}}}</script></span> of <var class="Arg">N</var>, and <code class="func">EI</code> returns <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msqrt></math>" style="position: relative;" tabindex="0" id="MathJax-Element-66-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.766em; display: inline-block;" role="math" id="MathJax-Span-634" class="math"><span style="display: inline-block; position: relative; width: 2.115em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.497em, 1002.15em, 2.739em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-635" class="mrow"><span id="MathJax-Span-636" class="msqrt"><span style="display: inline-block; position: relative; width: 2.114em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1001.41em, 4.15em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-637" class="mrow"><span id="MathJax-Span-638" class="texatom"><span id="MathJax-Span-639" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-640" class="mo">−</span><span id="MathJax-Span-641" class="texatom"><span id="MathJax-Span-642" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-643" class="mtext">N</span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1001.38em, 3.365em, -1000em); top: -3.981em; left: 0.737em;"><span style="display: inline-block; position: relative; width: 1.377em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0.877em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.99em; left: 0.426em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.858em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.248em; border-left: 0px solid; width: 0px; height: 1.365em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msqrt></math></span></span><script id="MathJax-Element-66" type="math/tex">\sqrt{{-\textit{N}}}</script></span>. By the chosen embedding of cyclotomic fields into the complex numbers, <code class="func">ER</code> returns the positive square root if <var class="Arg">N</var> is positive, and if <var class="Arg">N</var> is negative then <code class="code">ER(</code><var class="Arg">N</var><code class="code">) = EI(-</code><var class="Arg">N</var><code class="code">)</code> holds. In any case, <code class="code">EI(</code><var class="Arg">N</var><code class="code">) = E(4) * ER(</code><var class="Arg">N</var><code class="code">)</code>.</p>
-
-<p><code class="func">ER</code> is installed as method for the operation <code class="func">Sqrt</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap31_mj.html#X7E8F1FB87C229BB0"><span class="RefLink">31.12-5</span></a>), for rational argument.</p>
-
-<p>From a theorem of Gauss we know that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub><mo>=</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-67-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.641em; display: inline-block;" role="math" id="MathJax-Span-644" class="math"><span style="display: inline-block; position: relative; width: 2.019em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1001.97em, 2.949em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-645" class="mrow"><span id="MathJax-Span-646" class="msubsup"><span style="display: inline-block; position: relative; width: 1.047em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-647" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-648" class="texatom"><span id="MathJax-Span-649" class="mrow"><span id="MathJax-Span-650" class="texatom"><span id="MathJax-Span-651" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-652" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-653" class="mo">=</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.222em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub><mo>=</mo></math></span></span><script id="MathJax-Element-67" type="math/tex">b_{\textit{N}} =</script></span></p>
-
-<div class="pcenter"><table class="GAPDocTablenoborder">
-<tbody><tr>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mo>&#x2212;</mo><mn>1</mn><mo>+</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msqrt><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-68-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 6.766em; display: inline-block;" role="math" id="MathJax-Span-654" class="math"><span style="display: inline-block; position: relative; width: 5.192em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.689em, 1005.17em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-655" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-656" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-657" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-658" class="mn">1</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-659" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-660" class="msqrt"><span style="display: inline-block; position: relative; width: 1.429em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-661" class="mrow"><span id="MathJax-Span-662" class="texatom"><span id="MathJax-Span-663" class="mrow"><span id="MathJax-Span-664" class="texatom"><span id="MathJax-Span-665" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-666" class="mtext">N</span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1000.69em, 3.365em, -1000em); top: -3.981em; left: 0.737em;"><span style="display: inline-block; position: relative; width: 0.692em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0.192em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.858em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-667" class="mo">)</span><span id="MathJax-Span-668" class="texatom"><span id="MathJax-Span-669" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-670" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-671" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.409em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msqrt><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script id="MathJax-Element-68" type="math/tex">(-1 + \sqrt{{\textit{N}}}) / 2</script></span></td>
-<td class="tdcenter">if</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2261;</mo><mn>1</mn><mspace width="0.444em" /><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em" /><mn>4</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-69-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.328em; display: inline-block;" role="math" id="MathJax-Span-672" class="math"><span style="display: inline-block; position: relative; width: 6.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1006.35em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-673" class="mrow"><span id="MathJax-Span-674" class="texatom"><span id="MathJax-Span-675" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-676" class="mtext">N</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-677" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-678" class="mn">1</span><span id="MathJax-Span-679" class="TeXmathchoice"><span style="height: 0em; vertical-align: 0em; width: 0.5em; display: inline-block; overflow: hidden;" id="MathJax-Span-680" class="mspace"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-681" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-682" class="mi">mod</span><span style="height: 0em; vertical-align: 0em; width: 0.375em; display: inline-block; overflow: hidden;" id="MathJax-Span-683" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-684" class="mn">4</span><span style="font-family: STIXGeneral;" id="MathJax-Span-685" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>4</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-69" type="math/tex">\textit{N} \equiv 1 \pmod 4</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mo>&#x2212;</mo><mn>1</mn><mo>+</mo><mi>i</mi><msqrt><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msqrt><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-70-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.141em; display: inline-block;" role="math" id="MathJax-Span-686" class="math"><span style="display: inline-block; position: relative; width: 5.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.689em, 1005.46em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-687" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-688" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-689" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-690" class="mn">1</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-691" class="mo">+</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-692" class="mi">i</span><span id="MathJax-Span-693" class="msqrt"><span style="display: inline-block; position: relative; width: 1.429em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-694" class="mrow"><span id="MathJax-Span-695" class="texatom"><span id="MathJax-Span-696" class="mrow"><span id="MathJax-Span-697" class="texatom"><span id="MathJax-Span-698" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-699" class="mtext">N</span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1000.69em, 3.365em, -1000em); top: -3.981em; left: 0.737em;"><span style="display: inline-block; position: relative; width: 0.692em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0.192em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.858em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-700" class="mo">)</span><span id="MathJax-Span-701" class="texatom"><span id="MathJax-Span-702" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-703" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-704" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.409em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>+</mo><mi>i</mi><msqrt><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msqrt><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script id="MathJax-Element-70" type="math/tex">(-1 + i \sqrt{{\textit{N}}}) / 2</script></span></td>
-<td class="tdcenter">if</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>&#x2261;</mo><mo>&#x2212;</mo><mn>1</mn><mspace width="0.444em" /><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em" /><mn>4</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-71-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 9.203em; display: inline-block;" role="math" id="MathJax-Span-705" class="math"><span style="display: inline-block; position: relative; width: 7.067em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1007.02em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-706" class="mrow"><span id="MathJax-Span-707" class="texatom"><span id="MathJax-Span-708" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-709" class="mtext">N</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-710" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-711" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-712" class="mn">1</span><span id="MathJax-Span-713" class="TeXmathchoice"><span style="height: 0em; vertical-align: 0em; width: 0.5em; display: inline-block; overflow: hidden;" id="MathJax-Span-714" class="mspace"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-715" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-716" class="mi">mod</span><span style="height: 0em; vertical-align: 0em; width: 0.375em; display: inline-block; overflow: hidden;" id="MathJax-Span-717" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-718" class="mn">4</span><span style="font-family: STIXGeneral;" id="MathJax-Span-719" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo>≡</mo><mo>−</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>4</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-71" type="math/tex">\textit{N} \equiv -1 \pmod 4</script></span></td>
-</tr>
-</tbody></table><br><p> </p><br>
-</div>
-
-<p>So <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msqrt></math>" style="position: relative;" tabindex="0" id="MathJax-Element-72-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.891em; display: inline-block;" role="math" id="MathJax-Span-720" class="math"><span style="display: inline-block; position: relative; width: 1.442em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.497em, 1001.48em, 2.739em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-721" class="mrow"><span id="MathJax-Span-722" class="msqrt"><span style="display: inline-block; position: relative; width: 1.429em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-723" class="mrow"><span id="MathJax-Span-724" class="texatom"><span id="MathJax-Span-725" class="mrow"><span id="MathJax-Span-726" class="texatom"><span id="MathJax-Span-727" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-728" class="mtext">N</span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1000.69em, 3.365em, -1000em); top: -3.981em; left: 0.737em;"><span style="display: inline-block; position: relative; width: 0.692em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0.192em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.858em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.248em; border-left: 0px solid; width: 0px; height: 1.365em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msqrt></math></span></span><script id="MathJax-Element-72" type="math/tex">\sqrt{{\textit{N}}}</script></span> can be computed from <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-73-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-729" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1001.06em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-730" class="mrow"><span id="MathJax-Span-731" class="msubsup"><span style="display: inline-block; position: relative; width: 1.047em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-732" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-733" class="texatom"><span id="MathJax-Span-734" class="mrow"><span id="MathJax-Span-735" class="texatom"><span id="MathJax-Span-736" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-737" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.222em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-73" type="math/tex">b_{\textit{N}}</script></span>, see <code class="func">EB</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X844D62597D9EFE3C"><span class="RefLink">18.4-1</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">ER(3); EI(3);</span>
--E(12)^7+E(12)^11
-E(3)-E(3)^2
-</pre></div>
-
-<p><a id="X842338F184E63894" name="X842338F184E63894"></a></p>
-
-<h5>18.4-3 <span class="Heading">EY, EX, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2026;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-74-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-738" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.4em, 1000.9em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-739" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-740" class="mo">…</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.269em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>…</mo></math></span></span><script id="MathJax-Element-74" type="math/tex">\ldots</script></span>, ES</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EY</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EX</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EW</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EV</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EU</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ET</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ES</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>For the given integer <var class="Arg">N</var> <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&gt;</mo><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-75-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.953em; display: inline-block;" role="math" id="MathJax-Span-741" class="math"><span style="display: inline-block; position: relative; width: 1.49em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1001.46em, 2.812em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-742" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-743" class="mo">></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-744" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.094em; border-left: 0px solid; width: 0px; height: 1.035em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>2</mn></math></span></span><script id="MathJax-Element-75" type="math/tex">> 2</script></span>, let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mi>k</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-76-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-745" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1001.06em, 2.754em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-746" class="mrow"><span id="MathJax-Span-747" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-748" class="texatom"><span id="MathJax-Span-749" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-750" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-751" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.179em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mi>k</mi></msub></math></span></span><script id="MathJax-Element-76" type="math/tex">\textit{N}_k</script></span> denote the first integer with multiplicative order exactly <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-77-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-752" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.43em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-753" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-754" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script id="MathJax-Element-77" type="math/tex">k</script></span> modulo <var class="Arg">N</var>, chosen in the order of preference</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"></span><div style="text-align: center;" class="MathJax_Display"><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>1</mn><mo>,</mo><mo>&#x2212;</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>&#x2212;</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>&#x2212;</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mo>&#x2212;</mo><mn>4</mn><mo>,</mo><mo>&#x2026;</mo><mo>.</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-78-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 14.828em; display: inline-block;" role="math" id="MathJax-Span-755" class="math"><span style="display: inline-block; position: relative; width: 11.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1011.33em, 2.929em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-756" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-757" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-758" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-759" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-760" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-761" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-762" class="mn">2</span><span style="font-family: STIXGeneral;" id="MathJax-Span-763" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-764" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-765" class="mn">2</span><span style="font-family: STIXGeneral;" id="MathJax-Span-766" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-767" class="mn">3</span><span style="font-family: STIXGeneral;" id="MathJax-Span-768" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-769" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-770" class="mn">3</span><span style="font-family: STIXGeneral;" id="MathJax-Span-771" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-772" class="mn">4</span><span style="font-family: STIXGeneral;" id="MathJax-Span-773" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-774" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-775" class="mn">4</span><span style="font-family: STIXGeneral;" id="MathJax-Span-776" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-777" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-778" class="mo">.</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.246em; border-left: 0px solid; width: 0px; height: 1.187em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML MJX_Assistive_MathML_Block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>−</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mo>−</mo><mn>4</mn><mo>,</mo><mo>…</mo><mo>.</mo></math></span></span></div><script id="MathJax-Element-78" type="math/tex; mode=display">
-1, -1, 2, -2, 3, -3, 4, -4, \ldots .
-</script></p>
-
-<p>We define (with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>exp</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mn>2</mn><mi>&#x03C0;</mi><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-79-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.828em; display: inline-block;" role="math" id="MathJax-Span-779" class="math"><span style="display: inline-block; position: relative; width: 6.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1005.96em, 3.005em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-780" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-781" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-782" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-783" class="mi">exp</span><span id="MathJax-Span-784" class="mo"></span><span style="font-family: STIXGeneral;" id="MathJax-Span-785" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-786" class="mn">2</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-787" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-788" class="mi">i</span><span id="MathJax-Span-789" class="texatom"><span id="MathJax-Span-790" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-791" class="mo">/</span></span></span><span id="MathJax-Span-792" class="texatom"><span id="MathJax-Span-793" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-794" class="mtext">N</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-795" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.345em; border-left: 0px solid; width: 0px; height: 1.286em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>exp</mi><mo>â¡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>Ï€</mi><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-79" type="math/tex">z = \exp(2 \pi i/\textit{N})</script></span>)</p>
-
-<div class="pcenter"><table class="GAPDocTablenoborder">
-<tbody><tr>
-<td class="tdleft"><code class="code">EY(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-80-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-796" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.01em, 2.802em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-797" class="mrow"><span id="MathJax-Span-798" class="msubsup"><span style="display: inline-block; position: relative; width: 0.991em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.43em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-799" class="mi">y</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-800" class="texatom"><span id="MathJax-Span-801" class="mrow"><span id="MathJax-Span-802" class="texatom"><span id="MathJax-Span-803" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-804" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 0.966em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-80" type="math/tex">y_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-81-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.078em; display: inline-block;" role="math" id="MathJax-Span-805" class="math"><span style="display: inline-block; position: relative; width: 2.356em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.825em, 1002.36em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-806" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-807" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-808" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-809" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-810" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-811" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.107em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup></math></span></span><script id="MathJax-Element-81" type="math/tex">z + z^n</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>2</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-82-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-812" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-813" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-814" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-815" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-816" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-817" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-818" class="texatom"><span id="MathJax-Span-819" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-820" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-821" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-822" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-82" type="math/tex">(n = \textit{N}_2)</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EX(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-83-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-823" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.01em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-824" class="mrow"><span id="MathJax-Span-825" class="msubsup"><span style="display: inline-block; position: relative; width: 0.991em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-826" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-827" class="texatom"><span id="MathJax-Span-828" class="mrow"><span id="MathJax-Span-829" class="texatom"><span id="MathJax-Span-830" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-831" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 0.907em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-83" type="math/tex">x_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-84-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 6.078em; display: inline-block;" role="math" id="MathJax-Span-832" class="math"><span style="display: inline-block; position: relative; width: 4.663em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.543em, 1004.66em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-833" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-834" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-835" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-836" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-837" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-838" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-839" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-840" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-841" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-842" class="texatom"><span id="MathJax-Span-843" class="mrow"><span id="MathJax-Span-844" class="texatom"><span id="MathJax-Span-845" class="mrow"><span id="MathJax-Span-846" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-847" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-848" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.475em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup></math></span></span><script id="MathJax-Element-84" type="math/tex">z + z^n + z^{{n^2}}</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>3</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-85-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-849" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-850" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-851" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-852" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-853" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-854" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-855" class="texatom"><span id="MathJax-Span-856" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-857" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-858" class="mn">3</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-859" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-85" type="math/tex">(n = \textit{N}_3)</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EW</code>(<var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-86-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-860" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.2em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-861" class="mrow"><span id="MathJax-Span-862" class="msubsup"><span style="display: inline-block; position: relative; width: 1.214em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.65em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-863" class="mi">w</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span id="MathJax-Span-864" class="texatom"><span id="MathJax-Span-865" class="mrow"><span id="MathJax-Span-866" class="texatom"><span id="MathJax-Span-867" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-868" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 0.907em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-86" type="math/tex">w_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>3</mn></msup></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-87-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 9.078em; display: inline-block;" role="math" id="MathJax-Span-869" class="math"><span style="display: inline-block; position: relative; width: 6.971em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.543em, 1006.97em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-870" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-871" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-872" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-873" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-874" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-875" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-876" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-877" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-878" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-879" class="texatom"><span id="MathJax-Span-880" class="mrow"><span id="MathJax-Span-881" class="texatom"><span id="MathJax-Span-882" class="mrow"><span id="MathJax-Span-883" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-884" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-885" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-886" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-887" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-888" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-889" class="texatom"><span id="MathJax-Span-890" class="mrow"><span id="MathJax-Span-891" class="texatom"><span id="MathJax-Span-892" class="mrow"><span id="MathJax-Span-893" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-894" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-895" class="mn">3</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.475em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>3</mn></msup></mrow></mrow></msup></math></span></span><script id="MathJax-Element-87" type="math/tex">z + z^n + z^{{n^2}} + z^{{n^3}}</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>4</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-88-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-896" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-897" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-898" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-899" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-900" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-901" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-902" class="texatom"><span id="MathJax-Span-903" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-904" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-905" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-906" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>4</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-88" type="math/tex">(n = \textit{N}_4)</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EV(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-89-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-907" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.01em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-908" class="mrow"><span id="MathJax-Span-909" class="msubsup"><span style="display: inline-block; position: relative; width: 0.991em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.43em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-910" class="mi">v</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-911" class="texatom"><span id="MathJax-Span-912" class="mrow"><span id="MathJax-Span-913" class="texatom"><span id="MathJax-Span-914" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-915" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 0.907em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-89" type="math/tex">v_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>3</mn></msup></mrow></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>4</mn></msup></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-90-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 12.078em; display: inline-block;" role="math" id="MathJax-Span-916" class="math"><span style="display: inline-block; position: relative; width: 9.279em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.543em, 1009.28em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-917" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-918" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-919" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-920" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-921" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-922" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-923" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-924" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-925" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-926" class="texatom"><span id="MathJax-Span-927" class="mrow"><span id="MathJax-Span-928" class="texatom"><span id="MathJax-Span-929" class="mrow"><span id="MathJax-Span-930" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-931" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-932" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-933" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-934" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-935" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-936" class="texatom"><span id="MathJax-Span-937" class="mrow"><span id="MathJax-Span-938" class="texatom"><span id="MathJax-Span-939" class="mrow"><span id="MathJax-Span-940" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-941" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-942" class="mn">3</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-943" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-944" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-945" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-946" class="texatom"><span id="MathJax-Span-947" class="mrow"><span id="MathJax-Span-948" class="texatom"><span id="MathJax-Span-949" class="mrow"><span id="MathJax-Span-950" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-951" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-952" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.475em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>3</mn></msup></mrow></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>4</mn></msup></mrow></mrow></msup></math></span></span><script id="MathJax-Element-90" type="math/tex">z + z^n + z^{{n^2}} + z^{{n^3}} + z^{{n^4}}</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>5</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-91-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-953" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-954" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-955" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-956" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-957" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-958" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-959" class="texatom"><span id="MathJax-Span-960" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-961" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-962" class="mn">5</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-963" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>5</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-91" type="math/tex">(n = \textit{N}_5)</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EU(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-92-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-964" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.06em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-965" class="mrow"><span id="MathJax-Span-966" class="msubsup"><span style="display: inline-block; position: relative; width: 1.047em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-967" class="mi">u</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-968" class="texatom"><span id="MathJax-Span-969" class="mrow"><span id="MathJax-Span-970" class="texatom"><span id="MathJax-Span-971" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-972" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 0.907em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-92" type="math/tex">u_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>+</mo><mo>&#x2026;</mo><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>5</mn></msup></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-93-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 11.891em; display: inline-block;" role="math" id="MathJax-Span-973" class="math"><span style="display: inline-block; position: relative; width: 9.135em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.537em, 1009.14em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-974" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-975" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-976" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-977" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-978" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-979" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-980" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-981" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-982" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-983" class="texatom"><span id="MathJax-Span-984" class="mrow"><span id="MathJax-Span-985" class="texatom"><span id="MathJax-Span-986" class="mrow"><span id="MathJax-Span-987" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-988" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-989" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-990" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-991" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-992" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-993" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-994" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-995" class="texatom"><span id="MathJax-Span-996" class="mrow"><span id="MathJax-Span-997" class="texatom"><span id="MathJax-Span-998" class="mrow"><span id="MathJax-Span-999" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1000" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1001" class="mn">5</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.483em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>5</mn></msup></mrow></mrow></msup></math></span></span><script id="MathJax-Element-93" type="math/tex">z + z^n + z^{{n^2}} + \ldots + z^{{n^5}}</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>6</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-94-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-1002" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1003" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1004" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1005" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1006" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1007" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1008" class="texatom"><span id="MathJax-Span-1009" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1010" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1011" class="mn">6</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1012" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>6</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-94" type="math/tex">(n = \textit{N}_6)</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">ET(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-95-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.078em; display: inline-block;" role="math" id="MathJax-Span-1013" class="math"><span style="display: inline-block; position: relative; width: 0.817em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.762em, 1000.82em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1014" class="mrow"><span id="MathJax-Span-1015" class="msubsup"><span style="display: inline-block; position: relative; width: 0.825em; height: 0px;"><span style="position: absolute; clip: rect(3.3em, 1000.3em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1016" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span id="MathJax-Span-1017" class="texatom"><span id="MathJax-Span-1018" class="mrow"><span id="MathJax-Span-1019" class="texatom"><span id="MathJax-Span-1020" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1021" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.044em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-95" type="math/tex">t_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>+</mo><mo>&#x2026;</mo><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>6</mn></msup></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-96-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 11.891em; display: inline-block;" role="math" id="MathJax-Span-1022" class="math"><span style="display: inline-block; position: relative; width: 9.135em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.539em, 1009.14em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1023" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1024" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1025" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1026" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1027" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1028" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1029" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1030" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1031" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-1032" class="texatom"><span id="MathJax-Span-1033" class="mrow"><span id="MathJax-Span-1034" class="texatom"><span id="MathJax-Span-1035" class="mrow"><span id="MathJax-Span-1036" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1037" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1038" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1039" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1040" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1041" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1042" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1043" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-1044" class="texatom"><span id="MathJax-Span-1045" class="mrow"><span id="MathJax-Span-1046" class="texatom"><span id="MathJax-Span-1047" class="mrow"><span id="MathJax-Span-1048" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1049" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1050" class="mn">6</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.48em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>6</mn></msup></mrow></mrow></msup></math></span></span><script id="MathJax-Element-96" type="math/tex">z + z^n + z^{{n^2}} + \ldots + z^{{n^6}}</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>7</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-97-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-1051" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1052" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1053" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1054" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1055" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1056" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1057" class="texatom"><span id="MathJax-Span-1058" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1059" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1060" class="mn">7</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1061" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>7</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-97" type="math/tex">(n = \textit{N}_7)</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">ES(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-98-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.203em; display: inline-block;" role="math" id="MathJax-Span-1062" class="math"><span style="display: inline-block; position: relative; width: 0.913em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.866em, 1000.91em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1063" class="mrow"><span id="MathJax-Span-1064" class="msubsup"><span style="display: inline-block; position: relative; width: 0.936em; height: 0px;"><span style="position: absolute; clip: rect(3.404em, 1000.37em, 4.148em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1065" class="mi">s</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.389em;"><span id="MathJax-Span-1066" class="texatom"><span id="MathJax-Span-1067" class="mrow"><span id="MathJax-Span-1068" class="texatom"><span id="MathJax-Span-1069" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1070" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 0.908em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-98" type="math/tex">s_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>+</mo><mo>&#x2026;</mo><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>7</mn></msup></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-99-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 11.891em; display: inline-block;" role="math" id="MathJax-Span-1071" class="math"><span style="display: inline-block; position: relative; width: 9.135em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.543em, 1009.14em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1072" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1073" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1074" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1075" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1076" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1077" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1078" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1079" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1080" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-1081" class="texatom"><span id="MathJax-Span-1082" class="mrow"><span id="MathJax-Span-1083" class="texatom"><span id="MathJax-Span-1084" class="mrow"><span id="MathJax-Span-1085" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1086" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1087" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1088" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1089" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1090" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1091" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1092" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-1093" class="texatom"><span id="MathJax-Span-1094" class="mrow"><span id="MathJax-Span-1095" class="texatom"><span id="MathJax-Span-1096" class="mrow"><span id="MathJax-Span-1097" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1098" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1099" class="mn">7</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.475em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>+</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>7</mn></msup></mrow></mrow></msup></math></span></span><script id="MathJax-Element-99" type="math/tex">z + z^n + z^{{n^2}} + \ldots + z^{{n^7}}</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>8</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-100-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-1100" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1101" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1102" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1103" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1104" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1105" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1106" class="texatom"><span id="MathJax-Span-1107" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1108" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1109" class="mn">8</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1110" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>8</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-100" type="math/tex">(n = \textit{N}_8)</script></span></td>
-</tr>
-</tbody></table><br><p> </p><br>
-</div>
-
-<p>For the two-argument versions of the functions, see Section <code class="func">NK</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X844F0EBF849EDEB3"><span class="RefLink">18.4-5</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">EY(5);</span>
-E(5)+E(5)^4
-<span class="GAPprompt">gap></span> <span class="GAPinput">EW(16,3); EW(17,2);</span>
-0
-E(17)+E(17)^4+E(17)^13+E(17)^16
-</pre></div>
-
-<p><a id="X87CA3CA083D49AC7" name="X87CA3CA083D49AC7"></a></p>
-
-<h5>18.4-4 <span class="Heading">EM, EL, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2026;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-101-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-1111" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.4em, 1000.9em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1112" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1113" class="mo">…</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.269em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>…</mo></math></span></span><script id="MathJax-Element-101" type="math/tex">\ldots</script></span>, EJ</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EM</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EL</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EK</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ EJ</code>( <var class="Arg">N</var>[, <var class="Arg">d</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">N</var> be an integer, <var class="Arg">N</var> <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&gt;</mo><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-102-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.953em; display: inline-block;" role="math" id="MathJax-Span-1114" class="math"><span style="display: inline-block; position: relative; width: 1.49em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1001.46em, 2.812em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1115" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1116" class="mo">></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1117" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.094em; border-left: 0px solid; width: 0px; height: 1.035em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>2</mn></math></span></span><script id="MathJax-Element-102" type="math/tex">> 2</script></span>. We define (with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>exp</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mn>2</mn><mi>&#x03C0;</mi><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-103-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.828em; display: inline-block;" role="math" id="MathJax-Span-1118" class="math"><span style="display: inline-block; position: relative; width: 6.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1005.96em, 3.005em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1119" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1120" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1121" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1122" class="mi">exp</span><span id="MathJax-Span-1123" class="mo"></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1124" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1125" class="mn">2</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1126" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1127" class="mi">i</span><span id="MathJax-Span-1128" class="texatom"><span id="MathJax-Span-1129" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1130" class="mo">/</span></span></span><span id="MathJax-Span-1131" class="texatom"><span id="MathJax-Span-1132" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1133" class="mtext">N</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1134" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.345em; border-left: 0px solid; width: 0px; height: 1.286em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>exp</mi><mo>â¡</mo><mo stretchy="false">(</mo><mn>2</mn><mi>Ï€</mi><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-103" type="math/tex">z = \exp(2 \pi i/\textit{N})</script></span>)</p>
-
-<div class="pcenter"><table class="GAPDocTablenoborder">
-<tbody><tr>
-<td class="tdleft"><code class="code">EM(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-104-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.641em; display: inline-block;" role="math" id="MathJax-Span-1135" class="math"><span style="display: inline-block; position: relative; width: 1.25em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.25em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1136" class="mrow"><span id="MathJax-Span-1137" class="msubsup"><span style="display: inline-block; position: relative; width: 1.269em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.7em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1138" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span id="MathJax-Span-1139" class="texatom"><span id="MathJax-Span-1140" class="mrow"><span id="MathJax-Span-1141" class="texatom"><span id="MathJax-Span-1142" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1143" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 0.907em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-104" type="math/tex">m_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>&#x2212;</mo><msup><mi>z</mi><mi>n</mi></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-105-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.078em; display: inline-block;" role="math" id="MathJax-Span-1144" class="math"><span style="display: inline-block; position: relative; width: 2.356em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.825em, 1002.36em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1145" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1146" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1147" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-1148" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1149" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1150" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.107em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>−</mo><msup><mi>z</mi><mi>n</mi></msup></math></span></span><script id="MathJax-Element-105" type="math/tex">z - z^n</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>2</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-106-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-1151" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1152" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1153" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1154" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1155" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1156" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1157" class="texatom"><span id="MathJax-Span-1158" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1159" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1160" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1161" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-106" type="math/tex">(n = \textit{N}_2)</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EL(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-107-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.078em; display: inline-block;" role="math" id="MathJax-Span-1162" class="math"><span style="display: inline-block; position: relative; width: 0.817em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.82em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1163" class="mrow"><span id="MathJax-Span-1164" class="msubsup"><span style="display: inline-block; position: relative; width: 0.825em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1165" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span id="MathJax-Span-1166" class="texatom"><span id="MathJax-Span-1167" class="mrow"><span id="MathJax-Span-1168" class="texatom"><span id="MathJax-Span-1169" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1170" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.222em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-107" type="math/tex">l_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>&#x2212;</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>&#x2212;</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>3</mn></msup></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-108-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 9.078em; display: inline-block;" role="math" id="MathJax-Span-1171" class="math"><span style="display: inline-block; position: relative; width: 6.971em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.543em, 1006.97em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1172" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1173" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1174" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-1175" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1176" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1177" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1178" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1179" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1180" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-1181" class="texatom"><span id="MathJax-Span-1182" class="mrow"><span id="MathJax-Span-1183" class="texatom"><span id="MathJax-Span-1184" class="mrow"><span id="MathJax-Span-1185" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1186" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1187" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1188" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-1189" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1190" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-1191" class="texatom"><span id="MathJax-Span-1192" class="mrow"><span id="MathJax-Span-1193" class="texatom"><span id="MathJax-Span-1194" class="mrow"><span id="MathJax-Span-1195" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1196" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1197" class="mn">3</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.475em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>−</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>2</mn></msup></mrow></mrow></msup><mo>−</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>3</mn></msup></mrow></mrow></msup></math></span></span><script id="MathJax-Element-108" type="math/tex">z - z^n + z^{{n^2}} - z^{{n^3}}</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>4</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-109-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-1198" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1199" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1200" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1201" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1202" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1203" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1204" class="texatom"><span id="MathJax-Span-1205" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1206" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1207" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1208" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>4</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-109" type="math/tex">(n = \textit{N}_4)</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EK(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-110-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-1209" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1001.01em, 2.757em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1210" class="mrow"><span id="MathJax-Span-1211" class="msubsup"><span style="display: inline-block; position: relative; width: 0.991em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.46em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1212" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-1213" class="texatom"><span id="MathJax-Span-1214" class="mrow"><span id="MathJax-Span-1215" class="texatom"><span id="MathJax-Span-1216" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1217" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.222em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-110" type="math/tex">k_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>&#x2212;</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><mo>&#x2026;</mo><mo>&#x2212;</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>5</mn></msup></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-111-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.953em; display: inline-block;" role="math" id="MathJax-Span-1218" class="math"><span style="display: inline-block; position: relative; width: 6.875em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.537em, 1006.88em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1219" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1220" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1221" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-1222" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1223" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1224" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1225" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1226" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1227" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-1228" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1229" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-1230" class="texatom"><span id="MathJax-Span-1231" class="mrow"><span id="MathJax-Span-1232" class="texatom"><span id="MathJax-Span-1233" class="mrow"><span id="MathJax-Span-1234" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1235" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1236" class="mn">5</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.483em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>−</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><mo>…</mo><mo>−</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>5</mn></msup></mrow></mrow></msup></math></span></span><script id="MathJax-Element-111" type="math/tex">z - z^n + \ldots - z^{{n^5}}</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>6</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-112-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-1237" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1238" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1239" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1240" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1241" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1242" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1243" class="texatom"><span id="MathJax-Span-1244" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1245" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1246" class="mn">6</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1247" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>6</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-112" type="math/tex">(n = \textit{N}_6)</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EJ(</code><var class="Arg">N</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-113-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.078em; display: inline-block;" role="math" id="MathJax-Span-1248" class="math"><span style="display: inline-block; position: relative; width: 0.817em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.656em, 1000.82em, 2.803em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1249" class="mrow"><span id="MathJax-Span-1250" class="msubsup"><span style="display: inline-block; position: relative; width: 0.825em; height: 0px;"><span style="position: absolute; clip: rect(3.194em, 1000.28em, 4.342em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1251" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span id="MathJax-Span-1252" class="texatom"><span id="MathJax-Span-1253" class="mrow"><span id="MathJax-Span-1254" class="texatom"><span id="MathJax-Span-1255" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1256" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.242em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-113" type="math/tex">j_{\textit{N}}</script></span></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>&#x2212;</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><mo>&#x2026;</mo><mo>&#x2212;</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>7</mn></msup></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-114-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.953em; display: inline-block;" role="math" id="MathJax-Span-1257" class="math"><span style="display: inline-block; position: relative; width: 6.875em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.55em, 1006.88em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1258" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1259" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1260" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-1261" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1262" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1263" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1264" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1265" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1266" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-1267" class="msubsup"><span style="display: inline-block; position: relative; width: 1.121em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1268" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-1269" class="texatom"><span id="MathJax-Span-1270" class="mrow"><span id="MathJax-Span-1271" class="texatom"><span id="MathJax-Span-1272" class="mrow"><span id="MathJax-Span-1273" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1274" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.247em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-1275" class="mn">7</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.466em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>−</mo><msup><mi>z</mi><mi>n</mi></msup><mo>+</mo><mo>…</mo><mo>−</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>n</mi><mn>7</mn></msup></mrow></mrow></msup></math></span></span><script id="MathJax-Element-114" type="math/tex">z - z^n + \ldots - z^{{n^7}}</script></span></td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>8</mn></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-115-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.641em; display: inline-block;" role="math" id="MathJax-Span-1276" class="math"><span style="display: inline-block; position: relative; width: 3.558em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.51em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1277" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1278" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1279" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1280" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1281" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1282" class="texatom"><span id="MathJax-Span-1283" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1284" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1285" class="mn">8</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1286" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mn>8</mn></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-115" type="math/tex">(n = \textit{N}_8)</script></span></td>
-</tr>
-</tbody></table><br><p> </p><br>
-</div>
-
-<p>For the two-argument versions of the functions, see Section <code class="func">NK</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X844F0EBF849EDEB3"><span class="RefLink">18.4-5</span></a>).</p>
-
-<p><a id="X844F0EBF849EDEB3" name="X844F0EBF849EDEB3"></a></p>
-
-<h5>18.4-5 NK</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ NK</code>( <var class="Arg">N</var>, <var class="Arg">k</var>, <var class="Arg">d</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo stretchy="false">)</mo></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-116-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.078em; display: inline-block;" role="math" id="MathJax-Span-1287" class="math"><span style="display: inline-block; position: relative; width: 1.587em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.355em, 1001.59em, 2.922em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1288" class="mrow"><span id="MathJax-Span-1289" class="msubsup"><span style="display: inline-block; position: relative; width: 1.566em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1290" class="texatom"><span id="MathJax-Span-1291" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1292" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.9em, 4.26em, -1000em); top: -4.46em; left: 0.667em;"><span id="MathJax-Span-1293" class="texatom"><span id="MathJax-Span-1294" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1295" class="mo">(</span><span id="MathJax-Span-1296" class="texatom"><span id="MathJax-Span-1297" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1298" class="mtext">d</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1299" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.39em, 4.142em, -1000em); top: -3.672em; left: 0.667em;"><span id="MathJax-Span-1300" class="texatom"><span id="MathJax-Span-1301" class="mrow"><span id="MathJax-Span-1302" class="texatom"><span id="MathJax-Span-1303" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1304" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.487em; border-left: 0px solid; width: 0px; height: 1.787em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo stretchy="false">)</mo></mrow></msubsup></math></span></span><script id="MathJax-Element-116" type="math/tex">\textit{N}_{\textit{k}}^{(\textit{d})}</script></span> be the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-117-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.641em; display: inline-block;" role="math" id="MathJax-Span-1305" class="math"><span style="display: inline-block; position: relative; width: 2.788em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1002.74em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1306" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1307" class="mo">(</span><span id="MathJax-Span-1308" class="texatom"><span id="MathJax-Span-1309" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1310" class="mtext">d</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1311" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1312" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1313" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-117" type="math/tex">(\textit{d}+1)</script></span>-th integer with multiplicative order exactly <var class="Arg">k</var> modulo <var class="Arg">N</var>, chosen in the order of preference defined in Section <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X842338F184E63894"><span class="RefLink">18.4-3</span></a>; <code class="func">NK</code> returns <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo stretchy="false">)</mo></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-118-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.078em; display: inline-block;" role="math" id="MathJax-Span-1314" class="math"><span style="display: inline-block; position: relative; width: 1.587em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.355em, 1001.59em, 2.922em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1315" class="mrow"><span id="MathJax-Span-1316" class="msubsup"><span style="display: inline-block; position: relative; width: 1.566em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1317" class="texatom"><span id="MathJax-Span-1318" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1319" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.9em, 4.26em, -1000em); top: -4.46em; left: 0.667em;"><span id="MathJax-Span-1320" class="texatom"><span id="MathJax-Span-1321" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1322" class="mo">(</span><span id="MathJax-Span-1323" class="texatom"><span id="MathJax-Span-1324" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1325" class="mtext">d</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1326" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.39em, 4.142em, -1000em); top: -3.672em; left: 0.667em;"><span id="MathJax-Span-1327" class="texatom"><span id="MathJax-Span-1328" class="mrow"><span id="MathJax-Span-1329" class="texatom"><span id="MathJax-Span-1330" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1331" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.487em; border-left: 0px solid; width: 0px; height: 1.787em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo stretchy="false">)</mo></mrow></msubsup></math></span></span><script id="MathJax-Element-118" type="math/tex">\textit{N}_{\textit{k}}^{(\textit{d})}</script></span>; if there is no integer with the required multiplicative order, <code class="func">NK</code> returns <code class="keyw">fail</code>.</p>
-
-<p>We write <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow></msub><mo>=</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>,</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi></mrow></msubsup><mo>=</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>,</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi></mrow></msubsup><mo>=</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-119-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 16.641em; display: inline-block;" role="math" id="MathJax-Span-1332" class="math"><span style="display: inline-block; position: relative; width: 12.788em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.552em, 1012.79em, 3.115em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1333" class="mrow"><span id="MathJax-Span-1334" class="msubsup"><span style="display: inline-block; position: relative; width: 1.056em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1335" class="texatom"><span id="MathJax-Span-1336" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1337" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span id="MathJax-Span-1338" class="texatom"><span id="MathJax-Span-1339" class="mrow"><span id="MathJax-Span-1340" class="texatom"><span id="MathJax-Span-1341" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1342" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1343" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1344" class="msubsup"><span style="display: inline-block; position: relative; width: 1.566em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1345" class="texatom"><span id="MathJax-Span-1346" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1347" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.9em, 4.26em, -1000em); top: -4.46em; left: 0.667em;"><span id="MathJax-Span-1348" class="texatom"><span id="MathJax-Span-1349" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1350" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1351" class="mn">0</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1352" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.39em, 4.142em, -1000em); top: -3.672em; left: 0.667em;"><span id="MathJax-Span-1353" class="texatom"><span id="MathJax-Span-1354" class="mrow"><span id="MathJax-Span-1355" class="texatom"><span id="MathJax-Span-1356" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1357" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1358" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1359" class="msubsup"><span style="display: inline-block; position: relative; width: 1.056em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1360" class="texatom"><span id="MathJax-Span-1361" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1362" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.447em, 1000.32em, 4.115em, -1000em); top: -4.37em; left: 0.667em;"><span id="MathJax-Span-1363" class="texatom"><span id="MathJax-Span-1364" class="mrow"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-1365" class="mi">′</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.39em, 4.142em, -1000em); top: -3.727em; left: 0.667em;"><span id="MathJax-Span-1366" class="texatom"><span id="MathJax-Span-1367" class="mrow"><span id="MathJax-Span-1368" class="texatom"><span id="MathJax-Span-1369" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1370" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1371" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1372" class="msubsup"><span style="display: inline-block; position: relative; width: 1.566em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1373" class="texatom"><span id="MathJax-Span-1374" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1375" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.9em, 4.26em, -1000em); top: -4.46em; left: 0.667em;"><span id="MathJax-Span-1376" class="texatom"><span id="MathJax-Span-1377" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1378" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1379" class="mn">1</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1380" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.39em, 4.142em, -1000em); top: -3.672em; left: 0.667em;"><span id="MathJax-Span-1381" class="texatom"><span id="MathJax-Span-1382" class="mrow"><span id="MathJax-Span-1383" class="texatom"><span id="MathJax-Span-1384" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1385" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1386" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1387" class="msubsup"><span style="display: inline-block; position: relative; width: 1.223em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1388" class="texatom"><span id="MathJax-Span-1389" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1390" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.447em, 1000.56em, 4.115em, -1000em); top: -4.37em; left: 0.667em;"><span id="MathJax-Span-1391" class="texatom"><span id="MathJax-Span-1392" class="mrow"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-1393" class="mi">′</span><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-1394" class="mi">′</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.39em, 4.142em, -1000em); top: -3.727em; left: 0.667em;"><span id="MathJax-Span-1395" class="texatom"><span id="MathJax-Span-1396" class="mrow"><span id="MathJax-Span-1397" class="texatom"><span id="MathJax-Span-1398" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1399" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1400" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1401" class="msubsup"><span style="display: inline-block; position: relative; width: 1.566em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1402" class="texatom"><span id="MathJax-Span-1403" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1404" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.9em, 4.26em, -1000em); top: -4.46em; left: 0.667em;"><span id="MathJax-Span-1405" class="texatom"><span id="MathJax-Span-1406" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1407" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1408" class="mn">2</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1409" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.39em, 4.142em, -1000em); top: -3.672em; left: 0.667em;"><span id="MathJax-Span-1410" class="texatom"><span id="MathJax-Span-1411" class="mrow"><span id="MathJax-Span-1412" class="texatom"><span id="MathJax-Span-1413" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1414" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.487em; border-left: 0px solid; width: 0px; height: 1.781em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow></msub><mo>=</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>,</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">′</mi></mrow></msubsup><mo>=</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>,</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">′</mi><mi class="MJX-variant" mathvariant="normal">′</mi></mrow></msubsup><mo>=</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msubsup></math></span></span><script id="MathJax-Element-119" type="math/tex">\textit{N}_{\textit{k}} = \textit{N}_{\textit{k}}^{(0)}, \textit{N}_{\textit{k}}^{\prime} = \textit{N}_{\textit{k}}^{(1)}, \textit{N}_{\textit{k}}^{\prime\prime} = \textit{N}_{\textit{k}}^{(2)}</script></span> and so on.</p>
-
-<p>The algebraic numbers</p>
-
-<p class="center">\[
-y_{<var class="Arg">N</var>}^{\prime} = y_{<var class="Arg">N</var>}^{(1)},
-y_{<var class="Arg">N</var>}^{\prime\prime} = y_{<var class="Arg">N</var>}^{(2)}, \ldots,
-x_{<var class="Arg">N</var>}^{\prime}, x_{<var class="Arg">N</var>}^{\prime\prime}, \ldots,
-j_{<var class="Arg">N</var>}^{\prime}, j_{<var class="Arg">N</var>}^{\prime\prime}, \ldots
-\]</p>
-
-<p>are obtained on replacing <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-120-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-1415" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1001.06em, 2.754em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1416" class="mrow"><span id="MathJax-Span-1417" class="msubsup"><span style="display: inline-block; position: relative; width: 1.056em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1418" class="texatom"><span id="MathJax-Span-1419" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1420" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span id="MathJax-Span-1421" class="texatom"><span id="MathJax-Span-1422" class="mrow"><span id="MathJax-Span-1423" class="texatom"><span id="MathJax-Span-1424" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1425" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.179em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow></msub></math></span></span><script id="MathJax-Element-120" type="math/tex">\textit{N}_{\textit{k}}</script></span> in the definitions in the sections <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X842338F184E63894"><span class="RefLink">18.4-3</span></a> and <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X87CA3CA083D49AC7"><span class="RefLink">18.4-4</span></a> by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi></mrow></msubsup><mo>,</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi></mrow></msubsup><mo>,</mo><mo>&#x2026;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-121-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.391em; display: inline-block;" role="math" id="MathJax-Span-1426" class="math"><span style="display: inline-block; position: relative; width: 4.135em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.721em, 1004.02em, 3.059em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1427" class="mrow"><span id="MathJax-Span-1428" class="msubsup"><span style="display: inline-block; position: relative; width: 1.056em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1429" class="texatom"><span id="MathJax-Span-1430" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1431" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.447em, 1000.32em, 4.115em, -1000em); top: -4.37em; left: 0.667em;"><span id="MathJax-Span-1432" class="texatom"><span id="MathJax-Span-1433" class="mrow"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-1434" class="mi">′</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.39em, 4.142em, -1000em); top: -3.727em; left: 0.667em;"><span id="MathJax-Span-1435" class="texatom"><span id="MathJax-Span-1436" class="mrow"><span id="MathJax-Span-1437" class="texatom"><span id="MathJax-Span-1438" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1439" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1440" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1441" class="msubsup"><span style="display: inline-block; position: relative; width: 1.223em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1442" class="texatom"><span id="MathJax-Span-1443" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1444" class="mtext">N</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.447em, 1000.56em, 4.115em, -1000em); top: -4.37em; left: 0.667em;"><span id="MathJax-Span-1445" class="texatom"><span id="MathJax-Span-1446" class="mrow"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-1447" class="mi">′</span><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-1448" class="mi">′</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.39em, 4.142em, -1000em); top: -3.727em; left: 0.667em;"><span id="MathJax-Span-1449" class="texatom"><span id="MathJax-Span-1450" class="mrow"><span id="MathJax-Span-1451" class="texatom"><span id="MathJax-Span-1452" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1453" class="mtext">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1454" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1455" class="mo">…</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.415em; border-left: 0px solid; width: 0px; height: 1.491em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">′</mi></mrow></msubsup><mo>,</mo><msubsup><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">k</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">′</mi><mi class="MJX-variant" mathvariant="normal">′</mi></mrow></msubsup><mo>,</mo><mo>…</mo></math></span></span><script id="MathJax-Element-121" type="math/tex">\textit{N}_{\textit{k}}^{\prime}, \textit{N}_{\textit{k}}^{\prime\prime}, \ldots</script></span>; they can be entered as</p>
-
-<div class="pcenter"><table class="GAPDocTablenoborder">
-<tbody><tr>
-<td class="tdleft"><code class="code">EY(</code><var class="Arg">N</var>,<var class="Arg">d</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo stretchy="false">)</mo></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-122-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.766em; display: inline-block;" role="math" id="MathJax-Span-1456" class="math"><span style="display: inline-block; position: relative; width: 1.346em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.355em, 1001.35em, 2.904em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1457" class="mrow"><span id="MathJax-Span-1458" class="msubsup"><span style="display: inline-block; position: relative; width: 1.343em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.43em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1459" class="mi">y</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.9em, 4.26em, -1000em); top: -4.46em; left: 0.444em;"><span id="MathJax-Span-1460" class="texatom"><span id="MathJax-Span-1461" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1462" class="mo">(</span><span id="MathJax-Span-1463" class="texatom"><span id="MathJax-Span-1464" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1465" class="mtext">d</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1466" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.55em, 4.145em, -1000em); top: -3.693em; left: 0.444em;"><span id="MathJax-Span-1467" class="texatom"><span id="MathJax-Span-1468" class="mrow"><span id="MathJax-Span-1469" class="texatom"><span id="MathJax-Span-1470" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1471" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.463em; border-left: 0px solid; width: 0px; height: 1.764em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo stretchy="false">)</mo></mrow></msubsup></math></span></span><script id="MathJax-Element-122" type="math/tex">y_{\textit{N}}^{(\textit{d})}</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EX(</code><var class="Arg">N</var>,<var class="Arg">d</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo stretchy="false">)</mo></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-123-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.828em; display: inline-block;" role="math" id="MathJax-Span-1472" class="math"><span style="display: inline-block; position: relative; width: 1.394em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.355em, 1001.39em, 2.904em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1473" class="mrow"><span id="MathJax-Span-1474" class="msubsup"><span style="display: inline-block; position: relative; width: 1.397em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1475" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.9em, 4.26em, -1000em); top: -4.46em; left: 0.498em;"><span id="MathJax-Span-1476" class="texatom"><span id="MathJax-Span-1477" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1478" class="mo">(</span><span id="MathJax-Span-1479" class="texatom"><span id="MathJax-Span-1480" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1481" class="mtext">d</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1482" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.55em, 4.145em, -1000em); top: -3.693em; left: 0.444em;"><span id="MathJax-Span-1483" class="texatom"><span id="MathJax-Span-1484" class="mrow"><span id="MathJax-Span-1485" class="texatom"><span id="MathJax-Span-1486" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1487" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.463em; border-left: 0px solid; width: 0px; height: 1.764em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo stretchy="false">)</mo></mrow></msubsup></math></span></span><script id="MathJax-Element-123" type="math/tex">x_{\textit{N}}^{(\textit{d})}</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"></td>
-<td class="tdcenter"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2026;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-124-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-1488" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.4em, 1000.9em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1489" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1490" class="mo">…</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.269em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>…</mo></math></span></span><script id="MathJax-Element-124" type="math/tex">\ldots</script></span></td>
-<td class="tdleft"></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">EJ(</code><var class="Arg">N</var>,<var class="Arg">d</var><code class="code">)</code></td>
-<td class="tdcenter">=</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo stretchy="false">)</mo></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-125-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.641em; display: inline-block;" role="math" id="MathJax-Span-1491" class="math"><span style="display: inline-block; position: relative; width: 1.25em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.355em, 1001.25em, 2.904em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1492" class="mrow"><span id="MathJax-Span-1493" class="msubsup"><span style="display: inline-block; position: relative; width: 1.229em; height: 0px;"><span style="position: absolute; clip: rect(3.194em, 1000.28em, 4.342em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1494" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.9em, 4.26em, -1000em); top: -4.46em; left: 0.329em;"><span id="MathJax-Span-1495" class="texatom"><span id="MathJax-Span-1496" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1497" class="mo">(</span><span id="MathJax-Span-1498" class="texatom"><span id="MathJax-Span-1499" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1500" class="mtext">d</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1501" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.55em, 4.145em, -1000em); top: -3.693em; left: 0.278em;"><span id="MathJax-Span-1502" class="texatom"><span id="MathJax-Span-1503" class="mrow"><span id="MathJax-Span-1504" class="texatom"><span id="MathJax-Span-1505" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1506" class="mtext">N</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.463em; border-left: 0px solid; width: 0px; height: 1.764em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">N</mtext></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">d</mtext></mrow><mo stretchy="false">)</mo></mrow></msubsup></math></span></span><script id="MathJax-Element-125" type="math/tex">j_{\textit{N}}^{(\textit{d})}</script></span></td>
-</tr>
-</tbody></table><br><p> </p><br>
-</div>
-
-<p><a id="X812E334E7A869D33" name="X812E334E7A869D33"></a></p>
-
-<h5>18.4-6 AtlasIrrationality</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ AtlasIrrationality</code>( <var class="Arg">irratname</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">irratname</var> be a string that describes an
-irrational value as a linear combination in terms of the atomic
-irrationalities introduced in the sections <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X844D62597D9EFE3C"><span class="RefLink">18.4-1</span></a>, <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X813CF4327C4B4D29"><span class="RefLink">18.4-2</span></a>, <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X842338F184E63894"><span class="RefLink">18.4-3</span></a>, <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X87CA3CA083D49AC7"><span class="RefLink">18.4-4</span></a>. These irrational values are defined in <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCCN85">[CCNPW85, Chapter 6, Section 10]</a>, and the following description is mainly copied from there. If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>N</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-126-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.453em; display: inline-block;" role="math" id="MathJax-Span-1507" class="math"><span style="display: inline-block; position: relative; width: 1.106em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.11em, 2.805em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1508" class="mrow"><span id="MathJax-Span-1509" class="msubsup"><span style="display: inline-block; position: relative; width: 1.089em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1510" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1511" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.334em; border-left: 0px solid; width: 0px; height: 0.97em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>N</mi></msub></math></span></span><script id="MathJax-Element-126" type="math/tex">q_N</script></span> is such a value (e.g. <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>24</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-127-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.641em; display: inline-block;" role="math" id="MathJax-Span-1512" class="math"><span style="display: inline-block; position: relative; width: 1.25em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.584em, 1001.25em, 2.91em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1513" class="mrow"><span id="MathJax-Span-1514" class="msubsup"><span style="display: inline-block; position: relative; width: 1.226em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.43em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1515" class="mi">y</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.447em, 1000.56em, 4.115em, -1000em); top: -4.315em; left: 0.444em;"><span id="MathJax-Span-1516" class="texatom"><span id="MathJax-Span-1517" class="mrow"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-1518" class="mi">′</span><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-1519" class="mi">′</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.78em, 4.135em, -1000em); top: -3.677em; left: 0.444em;"><span id="MathJax-Span-1520" class="texatom"><span id="MathJax-Span-1521" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1522" class="mn">24</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.47em; border-left: 0px solid; width: 0px; height: 1.474em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mn>24</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">′</mi><mi class="MJX-variant" mathvariant="normal">′</mi></mrow></msubsup></math></span></span><script id="MathJax-Element-127" type="math/tex">y_{24}^{\prime\prime}</script></span>) then linear combinations of algebraic conjugates of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>N</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-128-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.453em; display: inline-block;" role="math" id="MathJax-Span-1523" class="math"><span style="display: inline-block; position: relative; width: 1.106em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.11em, 2.805em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1524" class="mrow"><span id="MathJax-Span-1525" class="msubsup"><span style="display: inline-block; position: relative; width: 1.089em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1526" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1527" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.334em; border-left: 0px solid; width: 0px; height: 0.97em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>N</mi></msub></math></span></span><script id="MathJax-Element-128" type="math/tex">q_N</script></span> are abbreviated as in the following examples:</p>
-
-<div class="pcenter"><table class="GAPDocTablenoborder">
-<tbody><tr>
-<td class="tdleft"><code class="code">2qN+3&5-4&7+&9</code></td>
-<td class="tdcenter">means</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi>q</mi><mi>N</mi></msub><mo>+</mo><mn>3</mn><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mn>5</mn></mrow></mrow></msubsup><mo>&#x2212;</mo><mn>4</mn><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mn>7</mn></mrow></mrow></msubsup><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mn>9</mn></mrow></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-129-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 12.953em; display: inline-block;" role="math" id="MathJax-Span-1528" class="math"><span style="display: inline-block; position: relative; width: 9.952em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.659em, 1009.95em, 3.096em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1529" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1530" class="mn">2</span><span id="MathJax-Span-1531" class="msubsup"><span style="display: inline-block; position: relative; width: 1.089em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1532" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1533" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1534" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1535" class="mn">3</span><span id="MathJax-Span-1536" class="msubsup"><span style="display: inline-block; position: relative; width: 1.298em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1537" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.36em, 1000.8em, 4.145em, -1000em); top: -4.345em; left: 0.5em;"><span id="MathJax-Span-1538" class="texatom"><span id="MathJax-Span-1539" class="mrow"><span id="MathJax-Span-1540" class="texatom"><span id="MathJax-Span-1541" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1542" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1543" class="mn">5</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1544" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1545" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1546" class="mn">4</span><span id="MathJax-Span-1547" class="msubsup"><span style="display: inline-block; position: relative; width: 1.298em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1548" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.378em, 1000.8em, 4.14em, -1000em); top: -4.34em; left: 0.5em;"><span id="MathJax-Span-1549" class="texatom"><span id="MathJax-Span-1550" class="mrow"><span id="MathJax-Span-1551" class="texatom"><span id="MathJax-Span-1552" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1553" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1554" class="mn">7</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1555" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1556" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1557" class="msubsup"><span style="display: inline-block; position: relative; width: 1.298em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1558" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.8em, 4.15em, -1000em); top: -4.35em; left: 0.5em;"><span id="MathJax-Span-1559" class="texatom"><span id="MathJax-Span-1560" class="mrow"><span id="MathJax-Span-1561" class="texatom"><span id="MathJax-Span-1562" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1563" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1564" class="mn">9</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1565" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.463em; border-left: 0px solid; width: 0px; height: 1.618em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi>q</mi><mi>N</mi></msub><mo>+</mo><mn>3</mn><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mn>5</mn></mrow></mrow></msubsup><mo>−</mo><mn>4</mn><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mn>7</mn></mrow></mrow></msubsup><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mn>9</mn></mrow></mrow></msubsup></math></span></span><script id="MathJax-Element-129" type="math/tex">2 q_N + 3 q_N^{{*5}} - 4 q_N^{{*7}} + q_N^{{*9}}</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">4qN&3&5&7-3&4</code></td>
-<td class="tdcenter">means</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo stretchy="false">(</mo><msub><mi>q</mi><mi>N</mi></msub><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mn>3</mn></mrow></mrow></msubsup><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mn>5</mn></mrow></mrow></msubsup><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mn>7</mn></mrow></mrow></msubsup><mo stretchy="false">)</mo><mo>&#x2212;</mo><mn>3</mn><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mn>11</mn></mrow></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-130-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 16.891em; display: inline-block;" role="math" id="MathJax-Span-1566" class="math"><span style="display: inline-block; position: relative; width: 12.981em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.659em, 1012.98em, 3.096em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1567" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1568" class="mn">4</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1569" class="mo">(</span><span id="MathJax-Span-1570" class="msubsup"><span style="display: inline-block; position: relative; width: 1.089em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1571" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1572" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1573" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1574" class="msubsup"><span style="display: inline-block; position: relative; width: 1.298em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1575" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.8em, 4.145em, -1000em); top: -4.345em; left: 0.5em;"><span id="MathJax-Span-1576" class="texatom"><span id="MathJax-Span-1577" class="mrow"><span id="MathJax-Span-1578" class="texatom"><span id="MathJax-Span-1579" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1580" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1581" class="mn">3</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1582" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1583" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1584" class="msubsup"><span style="display: inline-block; position: relative; width: 1.298em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1585" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.36em, 1000.8em, 4.145em, -1000em); top: -4.345em; left: 0.5em;"><span id="MathJax-Span-1586" class="texatom"><span id="MathJax-Span-1587" class="mrow"><span id="MathJax-Span-1588" class="texatom"><span id="MathJax-Span-1589" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1590" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1591" class="mn">5</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1592" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1593" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1594" class="msubsup"><span style="display: inline-block; position: relative; width: 1.298em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1595" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.378em, 1000.8em, 4.14em, -1000em); top: -4.34em; left: 0.5em;"><span id="MathJax-Span-1596" class="texatom"><span id="MathJax-Span-1597" class="mrow"><span id="MathJax-Span-1598" class="texatom"><span id="MathJax-Span-1599" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1600" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1601" class="mn">7</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1602" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1603" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1604" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1605" class="mn">3</span><span id="MathJax-Span-1606" class="msubsup"><span style="display: inline-block; position: relative; width: 1.652em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1607" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1001.15em, 4.135em, -1000em); top: -4.335em; left: 0.5em;"><span id="MathJax-Span-1608" class="texatom"><span id="MathJax-Span-1609" class="mrow"><span id="MathJax-Span-1610" class="texatom"><span id="MathJax-Span-1611" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1612" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1613" class="mn">11</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1614" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.463em; border-left: 0px solid; width: 0px; height: 1.618em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo stretchy="false">(</mo><msub><mi>q</mi><mi>N</mi></msub><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mn>3</mn></mrow></mrow></msubsup><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mn>5</mn></mrow></mrow></msubsup><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mn>7</mn></mrow></mrow></msubsup><mo stretchy="false">)</mo><mo>−</mo><mn>3</mn><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mn>11</mn></mrow></mrow></msubsup></math></span></span><script id="MathJax-Element-130" type="math/tex">4 (q_N + q_N^{{*3}} + q_N^{{*5}} + q_N^{{*7}}) - 3 q_N^{{*11}}</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><code class="code">4qN*3&5+&7</code></td>
-<td class="tdcenter">means</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo stretchy="false">(</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mn>3</mn></mrow></mrow></msubsup><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mn>5</mn></mrow></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mn>7</mn></mrow></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-131-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 9.641em; display: inline-block;" role="math" id="MathJax-Span-1615" class="math"><span style="display: inline-block; position: relative; width: 7.404em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.659em, 1007.4em, 3.096em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1616" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1617" class="mn">4</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1618" class="mo">(</span><span id="MathJax-Span-1619" class="msubsup"><span style="display: inline-block; position: relative; width: 1.298em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1620" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.8em, 4.145em, -1000em); top: -4.345em; left: 0.5em;"><span id="MathJax-Span-1621" class="texatom"><span id="MathJax-Span-1622" class="mrow"><span id="MathJax-Span-1623" class="texatom"><span id="MathJax-Span-1624" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1625" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1626" class="mn">3</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1627" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1628" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1629" class="msubsup"><span style="display: inline-block; position: relative; width: 1.298em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1630" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.36em, 1000.8em, 4.145em, -1000em); top: -4.345em; left: 0.5em;"><span id="MathJax-Span-1631" class="texatom"><span id="MathJax-Span-1632" class="mrow"><span id="MathJax-Span-1633" class="texatom"><span id="MathJax-Span-1634" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1635" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1636" class="mn">5</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1637" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1638" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1639" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1640" class="msubsup"><span style="display: inline-block; position: relative; width: 1.298em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1641" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.378em, 1000.8em, 4.14em, -1000em); top: -4.34em; left: 0.5em;"><span id="MathJax-Span-1642" class="texatom"><span id="MathJax-Span-1643" class="mrow"><span id="MathJax-Span-1644" class="texatom"><span id="MathJax-Span-1645" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1646" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1647" class="mn">7</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1648" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.463em; border-left: 0px solid; width: 0px; height: 1.618em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo stretchy="false">(</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mn>3</mn></mrow></mrow></msubsup><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mn>5</mn></mrow></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mn>7</mn></mrow></mrow></msubsup></math></span></span><script id="MathJax-Element-131" type="math/tex">4 (q_N^{{*3}} + q_N^{{*5}}) + q_N^{{*7}}</script></span></td>
-</tr>
-</tbody></table><br><p> </p><br>
-</div>
-
-<p>To explain the "ampersand" syntax in general we remark that "&k" is interpreted as <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mi>k</mi></mrow></mrow></msubsup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-132-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.641em; display: inline-block;" role="math" id="MathJax-Span-1649" class="math"><span style="display: inline-block; position: relative; width: 1.25em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.473em, 1001.25em, 2.904em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1650" class="mrow"><span id="MathJax-Span-1651" class="msubsup"><span style="display: inline-block; position: relative; width: 1.271em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1652" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.77em, 4.142em, -1000em); top: -4.343em; left: 0.5em;"><span id="MathJax-Span-1653" class="texatom"><span id="MathJax-Span-1654" class="mrow"><span id="MathJax-Span-1655" class="texatom"><span id="MathJax-Span-1656" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1657" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1658" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.59em, 4.145em, -1000em); top: -3.693em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1659" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.463em; border-left: 0px solid; width: 0px; height: 1.611em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>q</mi><mi>N</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mi>k</mi></mrow></mrow></msubsup></math></span></span><script id="MathJax-Element-132" type="math/tex">q_N^{{*k}}</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>N</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-133-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.453em; display: inline-block;" role="math" id="MathJax-Span-1660" class="math"><span style="display: inline-block; position: relative; width: 1.106em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.11em, 2.805em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1661" class="mrow"><span id="MathJax-Span-1662" class="msubsup"><span style="display: inline-block; position: relative; width: 1.089em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1663" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1664" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.334em; border-left: 0px solid; width: 0px; height: 0.97em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>N</mi></msub></math></span></span><script id="MathJax-Element-133" type="math/tex">q_N</script></span> is the most recently named atomic irrationality, and that the scope of any premultiplying coefficient is broken by a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-134-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-1665" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.953em, 1000.62em, 2.829em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1666" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1667" class="mo">+</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.116em; border-left: 0px solid; width: 0px; height: 0.889em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo></math></span></span><script id="MathJax-Element-134" type="math/tex">+</script></span> or <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2212;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-135-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-1668" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.214em, 1000.61em, 2.668em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1669" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1670" class="mo">−</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: 0.093em; border-left: 0px solid; width: 0px; height: 0.341em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo></math></span></span><script id="MathJax-Element-135" type="math/tex">-</script></span> sign, but not by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">&#x0026;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-136-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-1671" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.74em, 2.801em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1672" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1673" class="mi">&</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.079em; border-left: 0px solid; width: 0px; height: 1.021em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">&</mi></math></span></span><script id="MathJax-Element-136" type="math/tex">\&</script></span> or <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2217;</mo><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-137-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-1674" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.96em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1675" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1676" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1677" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∗</mo><mi>k</mi></math></span></span><script id="MathJax-Element-137" type="math/tex">*k</script></span>. The algebraic conjugations indicated by the ampersands apply directly to the <em>atomic</em> irrationality <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>N</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-138-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.453em; display: inline-block;" role="math" id="MathJax-Span-1678" class="math"><span style="display: inline-block; position: relative; width: 1.106em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.11em, 2.805em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1679" class="mrow"><span id="MathJax-Span-1680" class="msubsup"><span style="display: inline-block; position: relative; width: 1.089em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1681" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1682" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.334em; border-left: 0px solid; width: 0px; height: 0.97em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>N</mi></msub></math></span></span><script id="MathJax-Element-138" type="math/tex">q_N</script></span>, even when, as in the last example, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>N</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-139-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.453em; display: inline-block;" role="math" id="MathJax-Span-1683" class="math"><span style="display: inline-block; position: relative; width: 1.106em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.11em, 2.805em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1684" class="mrow"><span id="MathJax-Span-1685" class="msubsup"><span style="display: inline-block; position: relative; width: 1.089em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.344em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1686" class="mi">q</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1687" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.042em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.334em; border-left: 0px solid; width: 0px; height: 0.97em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>N</mi></msub></math></span></span><script id="MathJax-Element-139" type="math/tex">q_N</script></span> first appears with another conjugacy <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2217;</mo><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-140-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-1688" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.96em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1689" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1690" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1691" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∗</mo><mi>k</mi></math></span></span><script id="MathJax-Element-140" type="math/tex">*k</script></span>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "b7*3" );</span>
-E(7)^3+E(7)^5+E(7)^6
-<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "y'''24" );</span>
-E(24)-E(24)^19
-<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "-3y'''24*13&5" );</span>
-3*E(8)-3*E(8)^3
-<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "3y'''24*13-2&5" );</span>
--3*E(24)-2*E(24)^11+2*E(24)^17+3*E(24)^19
-<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "3y'''24*13-&5" );</span>
--3*E(24)-E(24)^11+E(24)^17+3*E(24)^19
-<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "3y'''24*13-4&5&7" );</span>
--7*E(24)-4*E(24)^11+4*E(24)^17+7*E(24)^19
-<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasIrrationality( "3y'''24&7" );</span>
-6*E(24)-6*E(24)^19
-</pre></div>
-
-<p><a id="X79FE34337DF2CD10" name="X79FE34337DF2CD10"></a></p>
-
-<h4>18.5 <span class="Heading">Galois Conjugacy of Cyclotomics</span></h4>
-
-<p><a id="X79EE9097783128C4" name="X79EE9097783128C4"></a></p>
-
-<h5>18.5-1 GaloisCyc</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ GaloisCyc</code>( <var class="Arg">cyc</var>, <var class="Arg">k</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ GaloisCyc</code>( <var class="Arg">list</var>, <var class="Arg">k</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>For a cyclotomic <var class="Arg">cyc</var> and an integer <var class="Arg">k</var>, <code class="func">GaloisCyc</code> returns the cyclotomic obtained by raising the roots of unity in the Zumbroich basis representation of <var class="Arg">cyc</var> to the <var class="Arg">k</var>-th power. If <var class="Arg">k</var> is coprime to the integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-141-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1692" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1693" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1694" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-141" type="math/tex">n</script></span>, <code class="code">GaloisCyc( ., <var class="Arg">k</var> )</code> acts as a Galois automorphism of the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-142-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1695" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1696" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1697" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-142" type="math/tex">n</script></span>-th cyclotomic field (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7E4AB4B17C7BA10C"><span class="RefLink">60.4</span></a>); to get the Galois automorphisms themselves, use <code class="func">GaloisGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html#X80CAA5BA82F09ED2"><span class="RefLink">58.3-1</span></a>).</p>
-
-<p>The <em>complex conjugate</em> of <var class="Arg">cyc</var> is <code class="code">GaloisCyc( <var class="Arg">cyc</var>, -1 )</code>, which can also be computed using <code class="func">ComplexConjugate</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7BE001A0811CD599"><span class="RefLink">18.5-2</span></a>).</p>
-
-<p>For a list or matrix <var class="Arg">list</var> of cyclotomics, <code class="func">GaloisCyc</code> returns the list obtained by applying <code class="func">GaloisCyc</code> to the entries of <var class="Arg">list</var>.</p>
-
-<p><a id="X7BE001A0811CD599" name="X7BE001A0811CD599"></a></p>
-
-<h5>18.5-2 ComplexConjugate</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ComplexConjugate</code>( <var class="Arg">z</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ RealPart</code>( <var class="Arg">z</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ImaginaryPart</code>( <var class="Arg">z</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For a cyclotomic number <var class="Arg">z</var>, <code class="func">ComplexConjugate</code> returns <code class="code">GaloisCyc( <var class="Arg">z</var>, -1 )</code>, see <code class="func">GaloisCyc</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X79EE9097783128C4"><span class="RefLink">18.5-1</span></a>). For a quaternion <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">z</mtext></mrow><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub><mi>e</mi><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub><mi>i</mi><mo>+</mo><msub><mi>c</mi><mn>3</mn></msub><mi>j</mi><mo>+</mo><msub><mi>c</mi><mn>4</mn></msub><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-143-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 13.203em; display: inline-block;" role="math" id="MathJax-Span-1698" class="math"><span style="display: inline-block; position: relative; width: 10.144em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1010.14em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1699" class="mrow"><span id="MathJax-Span-1700" class="texatom"><span id="MathJax-Span-1701" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1702" class="mtext">z</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1703" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1704" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1705" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1706" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1707" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1708" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1709" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1710" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1711" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1712" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1713" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1714" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1715" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1716" class="mn">3</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1717" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1718" class="mo">+</span><span style="padding-left: 0.25em;" id="MathJax-Span-1719" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1720" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1721" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1722" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.282em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">z</mtext></mrow><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub><mi>e</mi><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub><mi>i</mi><mo>+</mo><msub><mi>c</mi><mn>3</mn></msub><mi>j</mi><mo>+</mo><msub><mi>c</mi><mn>4</mn></msub><mi>k</mi></math></span></span><script id="MathJax-Element-143" type="math/tex">\textit{z} = c_1 e + c_2 i + c_3 j + c_4 k</script></span>, <code class="func">ComplexConjugate</code> returns <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mn>1</mn></msub><mi>e</mi><mo>&#x2212;</mo><msub><mi>c</mi><mn>2</mn></msub><mi>i</mi><mo>&#x2212;</mo><msub><mi>c</mi><mn>3</mn></msub><mi>j</mi><mo>&#x2212;</mo><msub><mi>c</mi><mn>4</mn></msub><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-144-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 11.016em; display: inline-block;" role="math" id="MathJax-Span-1723" class="math"><span style="display: inline-block; position: relative; width: 8.462em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1008.46em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1724" class="mrow"><span id="MathJax-Span-1725" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1726" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1727" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1728" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1729" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-1730" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1731" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1732" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1733" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1734" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-1735" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1736" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1737" class="mn">3</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1738" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1739" class="mo">−</span><span style="padding-left: 0.25em;" id="MathJax-Span-1740" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1741" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1742" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1743" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.282em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mn>1</mn></msub><mi>e</mi><mo>−</mo><msub><mi>c</mi><mn>2</mn></msub><mi>i</mi><mo>−</mo><msub><mi>c</mi><mn>3</mn></msub><mi>j</mi><mo>−</mo><msub><mi>c</mi><mn>4</mn></msub><mi>k</mi></math></span></span><script id="MathJax-Element-144" type="math/tex">c_1 e - c_2 i - c_3 j - c_4 k</script></span>, see <code class="func">IsQuaternion</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap62_mj.html#X82B3A9077D0CB453"><span class="RefLink">62.8-8</span></a>).</p>
-
-<p>When <code class="func">ComplexConjugate</code> is called with a list then the result is the list of return values of <code class="func">ComplexConjugate</code> for the list entries in the corresponding positions.</p>
-
-<p>When <code class="func">ComplexConjugate</code> is defined for an object <var class="Arg">z</var> then <code class="func">RealPart</code> and <code class="func">ImaginaryPart</code> return <code class="code">(<var class="Arg">z</var> + ComplexConjugate( <var class="Arg">z</var> )) / 2</code> and <code class="code">(<var class="Arg">z</var> - ComplexConjugate( <var class="Arg">z</var> )) / 2 i</code>, respectively, where <code class="code">i</code> denotes the corresponding imaginary unit.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5) + E(5)^4, 2 );</span>
-E(5)^2+E(5)^3
-<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5), -1 ); # the complex conjugate</span>
-E(5)^4
-<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5) + E(5)^4, -1 ); # this value is real</span>
-E(5)+E(5)^4
-<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(15) + E(15)^4, 3 );</span>
-E(5)+E(5)^4
-<span class="GAPprompt">gap></span> <span class="GAPinput">ComplexConjugate( E(7) );</span>
-E(7)^6
-</pre></div>
-
-<p><a id="X7E361C057E97CA66" name="X7E361C057E97CA66"></a></p>
-
-<h5>18.5-3 StarCyc</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ StarCyc</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>If the cyclotomic <var class="Arg">cyc</var> is an irrational element of a quadratic extension of the rationals then <code class="func">StarCyc</code> returns the unique Galois conjugate of <var class="Arg">cyc</var> that is different from <var class="Arg">cyc</var>, otherwise <code class="keyw">fail</code> is returned. In the first case, the return value is often called <var class="Arg">cyc</var><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2217;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-145-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1744" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.029em, 1000.46em, 2.755em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1745" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1746" class="mo">∗</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.02em; border-left: 0px solid; width: 0px; height: 0.694em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∗</mo></math></span></span><script id="MathJax-Element-145" type="math/tex">*</script></span> (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C1941F17BE9FC21"><span class="RefLink">71.13</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">StarCyc( EB(5) ); StarCyc( E(5) );</span>
-E(5)^2+E(5)^3
-fail
-</pre></div>
-
-<p><a id="X84438F867B0CC299" name="X84438F867B0CC299"></a></p>
-
-<h5>18.5-4 Quadratic</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Quadratic</code>( <var class="Arg">cyc</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">cyc</var> be a cyclotomic integer that lies in a quadratic extension field of the rationals. Then we have <var class="Arg">cyc</var><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msqrt><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>d</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-146-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.578em; display: inline-block;" role="math" id="MathJax-Span-1747" class="math"><span style="display: inline-block; position: relative; width: 5.817em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.792em, 1005.82em, 3.035em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1748" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1749" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1750" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1751" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1752" class="mo">+</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1753" class="mi">b</span><span id="MathJax-Span-1754" class="msqrt"><span style="display: inline-block; position: relative; width: 1.237em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.144em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-1755" class="mrow"><span id="MathJax-Span-1756" class="texatom"><span id="MathJax-Span-1757" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1758" class="mi">n</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1000.5em, 3.365em, -1000em); top: -3.878em; left: 0.737em;"><span style="font-family: STIXGeneral;">‾</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.755em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1759" class="mo">)</span><span id="MathJax-Span-1760" class="texatom"><span id="MathJax-Span-1761" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1762" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1763" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.382em; border-left: 0px solid; width: 0px; height: 1.365em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msqrt><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>d</mi></math></span></span><script id="MathJax-Element-146" type="math/tex"> = (a + b \sqrt{{n}}) / d</script></span>, for integers <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-147-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1764" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1765" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1766" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.713em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script id="MathJax-Element-147" type="math/tex">a</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-148-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1767" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.45em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1768" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1769" class="mi">b</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script id="MathJax-Element-148" type="math/tex">b</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-149-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1770" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1771" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1772" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-149" type="math/tex">n</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-150-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1773" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.53em, 2.801em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1774" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1775" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.079em; border-left: 0px solid; width: 0px; height: 1.03em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script id="MathJax-Element-150" type="math/tex">d</script></span>, such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-151-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1776" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.53em, 2.801em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1777" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1778" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.079em; border-left: 0px solid; width: 0px; height: 1.03em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script id="MathJax-Element-151" type="math/tex">d</script></span> is either <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-152-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1779" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.38em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1780" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1781" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script id="MathJax-Element-152" type="math/tex">1</script></span> or <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-153-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1782" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.46em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1783" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1784" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span></span><script id="MathJax-Element-153" type="math/tex">2</script></span>. In this case, <code class="func">Quadratic</code> returns a record with the components <code class="code">a</code>, <code class="code">b</code>, <code class="code">root</code>, <code class="code">d</code>, <code class="code">ATLAS</code>, and <code class="code">display</code>; the values of the first four are <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-154-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1785" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1786" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1787" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.713em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script id="MathJax-Element-154" type="math/tex">a</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-155-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1788" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.45em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1789" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1790" class="mi">b</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script id="MathJax-Element-155" type="math/tex">b</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-156-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1791" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1792" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1793" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-156" type="math/tex">n</script></span>, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-157-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1794" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.53em, 2.801em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1795" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1796" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.079em; border-left: 0px solid; width: 0px; height: 1.03em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script id="MathJax-Element-157" type="math/tex">d</script></span>, the <code class="code">ATLAS</code> value is a (not necessarily shortest) representation of <var class="Arg">cyc</var> in terms of the <strong class="pkg">Atlas</strong> irrationalities <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-158-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-1797" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1001.2em, 2.756em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1798" class="mrow"><span id="MathJax-Span-1799" class="msubsup"><span style="display: inline-block; position: relative; width: 1.211em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1800" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-1801" class="texatom"><span id="MathJax-Span-1802" class="mrow"><span id="MathJax-Span-1803" class="texatom"><span id="MathJax-Span-1804" class="mrow"><span id="MathJax-Span-1805" class="texatom"><span id="MathJax-Span-1806" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1807" class="mo">|</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1808" class="mi">n</span><span id="MathJax-Span-1809" class="texatom"><span id="MathJax-Span-1810" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1811" class="mo">|</span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.27em; border-left: 0px solid; width: 0px; height: 1.221em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mrow></msub></math></span></span><script id="MathJax-Element-158" type="math/tex">b_{{|n|}}</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-159-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-1812" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.654em, 1001.01em, 2.756em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1813" class="mrow"><span id="MathJax-Span-1814" class="msubsup"><span style="display: inline-block; position: relative; width: 0.989em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.26em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1815" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span id="MathJax-Span-1816" class="texatom"><span id="MathJax-Span-1817" class="mrow"><span id="MathJax-Span-1818" class="texatom"><span id="MathJax-Span-1819" class="mrow"><span id="MathJax-Span-1820" class="texatom"><span id="MathJax-Span-1821" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1822" class="mo">|</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1823" class="mi">n</span><span id="MathJax-Span-1824" class="texatom"><span id="MathJax-Span-1825" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1826" class="mo">|</span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.27em; border-left: 0px solid; width: 0px; height: 1.183em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mrow></msub></math></span></span><script id="MathJax-Element-159" type="math/tex">i_{{|n|}}</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-160-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.453em; display: inline-block;" role="math" id="MathJax-Span-1827" class="math"><span style="display: inline-block; position: relative; width: 1.106em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.11em, 2.756em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1828" class="mrow"><span id="MathJax-Span-1829" class="msubsup"><span style="display: inline-block; position: relative; width: 1.1em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1830" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.389em;"><span id="MathJax-Span-1831" class="texatom"><span id="MathJax-Span-1832" class="mrow"><span id="MathJax-Span-1833" class="texatom"><span id="MathJax-Span-1834" class="mrow"><span id="MathJax-Span-1835" class="texatom"><span id="MathJax-Span-1836" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1837" class="mo">|</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1838" class="mi">n</span><span id="MathJax-Span-1839" class="texatom"><span id="MathJax-Span-1840" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1841" class="mo">|</span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.27em; border-left: 0px solid; width: 0px; height: 0.906em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mrow></msub></math></span></span><script id="MathJax-Element-160" type="math/tex">r_{{|n|}}</script></span>, and the <code class="code">display</code> value is a string that expresses <var class="Arg">cyc</var> in <strong class="pkg">GAP</strong> notation, corresponding to the value of the <code class="code">ATLAS</code> component.</p>
-
-<p>If <var class="Arg">cyc</var> is not a cyclotomic integer or does not lie in a quadratic extension field of the rationals then <code class="keyw">fail</code> is returned.</p>
-
-<p>If the denominator <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-161-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1842" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.53em, 2.801em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1843" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1844" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.079em; border-left: 0px solid; width: 0px; height: 1.03em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script id="MathJax-Element-161" type="math/tex">d</script></span> is <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-162-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1845" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.46em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1846" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1847" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span></span><script id="MathJax-Element-162" type="math/tex">2</script></span> then necessarily <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-163-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1848" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1849" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1850" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-163" type="math/tex">n</script></span> is congruent to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-164-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1851" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.38em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1852" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1853" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script id="MathJax-Element-164" type="math/tex">1</script></span> modulo <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-165-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1854" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.45em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1855" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1856" class="mn">4</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span></span><script id="MathJax-Element-165" type="math/tex">4</script></span>, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-166-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.078em; display: inline-block;" role="math" id="MathJax-Span-1857" class="math"><span style="display: inline-block; position: relative; width: 0.817em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1000.82em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1858" class="mrow"><span id="MathJax-Span-1859" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1860" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1861" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 0.902em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>n</mi></msub></math></span></span><script id="MathJax-Element-166" type="math/tex">r_n</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>i</mi><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-167-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1862" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.654em, 1000.72em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1863" class="mrow"><span id="MathJax-Span-1864" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.26em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1865" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1866" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.178em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>i</mi><mi>n</mi></msub></math></span></span><script id="MathJax-Element-167" type="math/tex">i_n</script></span> are not possible; we have <code class="code"><var class="Arg">cyc</var> = x + y * EB( root )</code> with <code class="code">y = b</code>, <code class="code">x = ( a + b ) / 2</code>.</p>
-
-<p>If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-168-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.016em; display: inline-block;" role="math" id="MathJax-Span-1867" class="math"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1002.2em, 2.801em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1868" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1869" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1870" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1871" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.079em; border-left: 0px solid; width: 0px; height: 1.03em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>1</mn></math></span></span><script id="MathJax-Element-168" type="math/tex">d = 1</script></span>, we have the possibilities <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-169-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.328em; display: inline-block;" role="math" id="MathJax-Span-1872" class="math"><span style="display: inline-block; position: relative; width: 1.01em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.654em, 1001.01em, 2.756em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1873" class="mrow"><span id="MathJax-Span-1874" class="msubsup"><span style="display: inline-block; position: relative; width: 0.989em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.26em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1875" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span id="MathJax-Span-1876" class="texatom"><span id="MathJax-Span-1877" class="mrow"><span id="MathJax-Span-1878" class="texatom"><span id="MathJax-Span-1879" class="mrow"><span id="MathJax-Span-1880" class="texatom"><span id="MathJax-Span-1881" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1882" class="mo">|</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1883" class="mi">n</span><span id="MathJax-Span-1884" class="texatom"><span id="MathJax-Span-1885" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1886" class="mo">|</span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.27em; border-left: 0px solid; width: 0px; height: 1.183em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>i</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mrow></msub></math></span></span><script id="MathJax-Element-169" type="math/tex">i_{{|n|}}</script></span> for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&lt;</mo><mo>&#x2212;</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-170-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.828em; display: inline-block;" role="math" id="MathJax-Span-1887" class="math"><span style="display: inline-block; position: relative; width: 2.933em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.83em, 2.812em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1888" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1889" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1890" class="mo"><</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1891" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1892" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.094em; border-left: 0px solid; width: 0px; height: 1.035em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo><</mo><mo>−</mo><mn>1</mn></math></span></span><script id="MathJax-Element-170" type="math/tex">n < -1</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>&#x2217;</mo><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-171-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.516em; display: inline-block;" role="math" id="MathJax-Span-1893" class="math"><span style="display: inline-block; position: relative; width: 3.462em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1003.45em, 2.829em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1894" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1895" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1896" class="mo">+</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1897" class="mi">b</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1898" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1899" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.116em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>∗</mo><mi>i</mi></math></span></span><script id="MathJax-Element-171" type="math/tex">a + b * i</script></span> for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo>&#x2212;</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-172-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.828em; display: inline-block;" role="math" id="MathJax-Span-1900" class="math"><span style="display: inline-block; position: relative; width: 2.933em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.83em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1901" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1902" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1903" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1904" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1905" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 1.016em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo>−</mo><mn>1</mn></math></span></span><script id="MathJax-Element-172" type="math/tex">n = -1</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>&#x2217;</mo><msub><mi>r</mi><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-173-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.203em; display: inline-block;" role="math" id="MathJax-Span-1906" class="math"><span style="display: inline-block; position: relative; width: 3.99em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1003.99em, 2.945em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1907" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1908" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1909" class="mo">+</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1910" class="mi">b</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1911" class="mo">∗</span><span style="padding-left: 0.25em;" id="MathJax-Span-1912" class="msubsup"><span style="display: inline-block; position: relative; width: 0.818em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1913" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1914" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.216em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>∗</mo><msub><mi>r</mi><mi>n</mi></msub></math></span></span><script id="MathJax-Element-173" type="math/tex">a + b * r_n</script></span> for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&gt;</mo><mn>0</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-174-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.953em; display: inline-block;" role="math" id="MathJax-Span-1915" class="math"><span style="display: inline-block; position: relative; width: 2.26em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.24em, 2.812em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1916" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1917" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1918" class="mo">></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1919" class="mn">0</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.094em; border-left: 0px solid; width: 0px; height: 1.035em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>0</mn></math></span></span><script id="MathJax-Element-174" type="math/tex">n > 0</script></span>. Furthermore if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-175-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1920" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1921" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1922" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-175" type="math/tex">n</script></span> is congruent to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-176-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1923" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.38em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1924" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1925" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script id="MathJax-Element-176" type="math/tex">1</script></span> modulo <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-177-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1926" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.45em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1927" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1928" class="mn">4</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span></span><script id="MathJax-Element-177" type="math/tex">4</script></span>, also <var class="Arg">cyc</var> <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mo>&#x2217;</mo><mi>b</mi><mo>&#x2217;</mo><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-178-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 11.953em; display: inline-block;" role="math" id="MathJax-Span-1929" class="math"><span style="display: inline-block; position: relative; width: 9.183em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1009.18em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1930" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1931" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1932" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1933" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1934" class="mo">+</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1935" class="mi">b</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1936" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1937" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1938" class="mn">2</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1939" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1940" class="mi">b</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1941" class="mo">∗</span><span style="padding-left: 0.25em;" id="MathJax-Span-1942" class="msubsup"><span style="display: inline-block; position: relative; width: 1.211em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1943" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-1944" class="texatom"><span id="MathJax-Span-1945" class="mrow"><span id="MathJax-Span-1946" class="texatom"><span id="MathJax-Span-1947" class="mrow"><span id="MathJax-Span-1948" class="texatom"><span id="MathJax-Span-1949" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1950" class="mo">|</span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1951" class="mi">n</span><span id="MathJax-Span-1952" class="texatom"><span id="MathJax-Span-1953" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1954" class="mo">|</span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mo>∗</mo><mi>b</mi><mo>∗</mo><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mrow></msub></math></span></span><script id="MathJax-Element-178" type="math/tex">= (a+b) + 2 * b * b_{{|n|}}</script></span> is possible; the shortest string of these is taken as the value for the component <code class="code">ATLAS</code>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">Quadratic( EB(5) ); Quadratic( EB(27) );</span>
-rec( ATLAS := "b5", a := -1, b := 1, d := 2,
- display := "(-1+Sqrt(5))/2", root := 5 )
-rec( ATLAS := "1+3b3", a := -1, b := 3, d := 2,
- display := "(-1+3*Sqrt(-3))/2", root := -3 )
-<span class="GAPprompt">gap></span> <span class="GAPinput">Quadratic(0); Quadratic( E(5) );</span>
-rec( ATLAS := "0", a := 0, b := 0, d := 1, display := "0", root := 1 )
-fail
-</pre></div>
-
-<p><a id="X7DDDEC3F80543B7D" name="X7DDDEC3F80543B7D"></a></p>
-
-<h5>18.5-5 GaloisMat</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ GaloisMat</code>( <var class="Arg">mat</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">mat</var> be a matrix of cyclotomics. <code class="func">GaloisMat</code> calculates the complete orbits under the operation of the Galois group of the (irrational) entries of <var class="Arg">mat</var>, and the permutations of rows corresponding to the generators of the Galois group.</p>
-
-<p>If some rows of <var class="Arg">mat</var> are identical, only the first one is considered for the permutations, and a warning will be printed.</p>
-
-<p><code class="func">GaloisMat</code> returns a record with the components <code class="code">mat</code>, <code class="code">galoisfams</code>, and <code class="code">generators</code>.</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code">mat</code></strong></dt>
-<dd><p>a list with initial segment being the rows of <var class="Arg">mat</var> (<em>not</em> shallow copies of these rows); the list consists of full orbits under the action of the Galois group of the entries of <var class="Arg">mat</var> defined above. The last rows in the list are those not contained in <var class="Arg">mat</var> but must be added in order to complete the orbits; so if the orbits were already complete, <var class="Arg">mat</var> and <code class="code">mat</code> have identical rows.</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">galoisfams</code></strong></dt>
-<dd><p>a list that has the same length as the <code class="code">mat</code> component, its entries are either 1, 0, -1, or lists.</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code">galoisfams[i] = 1</code></strong></dt>
-<dd><p>means that <code class="code">mat[i]</code> consists of rationals, i.e., <code class="code">[ mat[i] ]</code> forms an orbit;</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">galoisfams[i] = -1</code></strong></dt>
-<dd><p>means that <code class="code">mat[i]</code> contains unknowns (see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap74_mj.html#X7C1FAB6280A02CCB"><span class="RefLink">74</span></a>); in this case <code class="code">[ mat[i] ]</code> is regarded as an orbit, too, even if <code class="code">mat[i]</code> contains irrational entries;</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">galoisfams[i] = </code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>l</mi><mn>1</mn></msub><mo>,</mo><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-179-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.266em; display: inline-block;" role="math" id="MathJax-Span-1955" class="math"><span style="display: inline-block; position: relative; width: 2.5em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1002.41em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1956" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1957" class="mo">[</span><span id="MathJax-Span-1958" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1959" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1960" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1961" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1962" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1963" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1964" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1965" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.216em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>l</mi><mn>1</mn></msub><mo>,</mo><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-179" type="math/tex">[ l_1, l_2 ]</script></span></strong></dt>
-<dd><p>(a list) means that <code class="code">mat[i]</code> is the first element of its orbit in <code class="code">mat</code>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-180-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1966" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.72em, 2.746em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1967" class="mrow"><span id="MathJax-Span-1968" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1969" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1970" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.208em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math></span></span><script id="MathJax-Element-180" type="math/tex">l_1</script></span> is the list of positions of rows that form the orbit, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-181-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1971" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.72em, 2.746em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1972" class="mrow"><span id="MathJax-Span-1973" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1974" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1975" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.208em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math></span></span><script id="MathJax-Element-181" type="math/tex">l_2</script></span> is the list of corresponding Galois automorphisms (as exponents, not as functions); so we have <code class="code">mat</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>=</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-182-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.828em; display: inline-block;" role="math" id="MathJax-Span-1976" class="math"><span style="display: inline-block; position: relative; width: 4.471em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1004.42em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1977" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1978" class="mo">[</span><span id="MathJax-Span-1979" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1980" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1981" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1982" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1983" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1984" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1985" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1986" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1987" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1988" class="mo">]</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1989" class="mo">=</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.282em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>=</mo></math></span></span><script id="MathJax-Element-182" type="math/tex">[ l_1[j] ][k] = </script></span> <code class="code">GaloisCyc( mat</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>,</mo><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-183-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.453em; display: inline-block;" role="math" id="MathJax-Span-1990" class="math"><span style="display: inline-block; position: relative; width: 4.183em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1004.09em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1991" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1992" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1993" class="mi">i</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1994" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1995" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1996" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1997" class="mo">]</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1998" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1999" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2000" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2001" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2002" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2003" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2004" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.282em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>,</mo><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-183" type="math/tex">[i][k], l_2[j]</script></span><code class="code"> )</code>;</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">galoisfams[i] = 0</code></strong></dt>
-<dd><p>means that <code class="code">mat[i]</code> is an element of a nontrivial orbit but not the first element of it.</p>
-
-</dd>
-</dl>
-</dd>
-<dt><strong class="Mark"><code class="code">generators</code></strong></dt>
-<dd><p>a list of permutations generating the permutation group corresponding to the action of the Galois group on the rows of <code class="code">mat</code>.</p>
-
-</dd>
-</dl>
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">GaloisMat( [ [ E(3), E(4) ] ] );</span>
-rec( galoisfams := [ [ [ 1, 2, 3, 4 ], [ 1, 7, 5, 11 ] ], 0, 0, 0 ],
- generators := [ (1,2)(3,4), (1,3)(2,4) ],
- mat := [ [ E(3), E(4) ], [ E(3), -E(4) ], [ E(3)^2, E(4) ],
- [ E(3)^2, -E(4) ] ] )
-<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisMat( [ [ 1, 1, 1 ], [ 1, E(3), E(3)^2 ] ] );</span>
-rec( galoisfams := [ 1, [ [ 2, 3 ], [ 1, 2 ] ], 0 ],
- generators := [ (2,3) ],
- mat := [ [ 1, 1, 1 ], [ 1, E(3), E(3)^2 ], [ 1, E(3)^2, E(3) ] ] )
-</pre></div>
-
-<p><a id="X7BB9F5957AA8C082" name="X7BB9F5957AA8C082"></a></p>
-
-<h5>18.5-6 RationalizedMat</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ RationalizedMat</code>( <var class="Arg">mat</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>returns the list of rationalized rows of <var class="Arg">mat</var>,
-which must be a matrix of cyclotomics. This is the set of sums over
-orbits under the action of the Galois group of the entries of <var class="Arg">mat</var> (see <code class="func">GaloisMat</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7DDDEC3F80543B7D"><span class="RefLink">18.5-5</span></a>)), so the operation may be viewed as a kind of trace on the rows.</p>
-
-<p>Note that no two rows of <var class="Arg">mat</var> should be equal.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">mat:= [ [ 1, 1, 1 ], [ 1, E(3), E(3)^2 ], [ 1, E(3)^2, E(3) ] ];;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">RationalizedMat( mat );</span>
-[ [ 1, 1, 1 ], [ 2, -1, -1 ] ]
-</pre></div>
-
-<p><a id="X8557FC2D7ACD6105" name="X8557FC2D7ACD6105"></a></p>
-
-<h4>18.6 <span class="Heading">Internally Represented Cyclotomics</span></h4>
-
-<p>The implementation of an <em>internally represented cyclotomic</em> is based on a list of length equal to its conductor. This means that the internal representation of a cyclotomic does <em>not</em> refer to the smallest number field but the smallest <em>cyclotomic</em>
- field containing it. The reason for this is the wish to reflect the
-natural embedding of two cyclotomic fields into a larger one that
-contains both. With such embeddings, it is easy to construct the sum or
-the product of two arbitrary cyclotomics (in possibly different fields)
-as an element of a cyclotomic field.</p>
-
-<p>The disadvantage of this approach is that the arithmetical operations
- are quite expensive, so the use of internally represented cyclotomics
-is not recommended for doing arithmetics over number fields, such as
-calculations with matrices of cyclotomics. But internally represented
-cyclotomics are good enough for dealing with irrationalities in
-character tables (see chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7B7A9EE881E01C10"><span class="RefLink">71</span></a>).</p>
-
-<p>For the representation of cyclotomics one has to recall that the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-184-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-2005" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2006" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2007" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-184" type="math/tex">n</script></span>-th cyclotomic field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mo stretchy="false">(</mo><msub><mi>e</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-185-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.016em; display: inline-block;" role="math" id="MathJax-Span-2008" class="math"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.26em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2009" class="mrow"><span id="MathJax-Span-2010" class="texatom"><span id="MathJax-Span-2011" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2012" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2013" class="mo">(</span><span id="MathJax-Span-2014" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2015" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2016" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2017" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mo stretchy="false">(</mo><msub><mi>e</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-185" type="math/tex">ℚ(e_n)</script></span> is a vector space of dimension <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C6;</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-186-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.328em; display: inline-block;" role="math" id="MathJax-Span-2018" class="math"><span style="display: inline-block; position: relative; width: 1.779em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1001.73em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2019" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2020" class="mi">φ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2021" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2022" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2023" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.27em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-186" type="math/tex">\varphi(n)</script></span> over the rationals where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C6;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-187-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.828em; display: inline-block;" role="math" id="MathJax-Span-2024" class="math"><span style="display: inline-block; position: relative; width: 0.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.59em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2025" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2026" class="mi">φ</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi></math></span></span><script id="MathJax-Element-187" type="math/tex">\varphi</script></span> denotes Euler's phi-function (see <code class="func">Phi</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap15_mj.html#X85A0C67982D9057A"><span class="RefLink">15.2-2</span></a>)).</p>
-
-<p>A special integral basis of cyclotomic fields is chosen that allows
-one to easily convert arbitrary sums of roots of unity into the basis,
-as well as to convert a cyclotomic represented w.r.t. the basis
-into the smallest possible cyclotomic field. This basis is accessible in
- <strong class="pkg">GAP</strong>, see <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a> for more information and references.</p>
-
-<p>Note that the set of all <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-188-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-2027" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2028" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2029" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-188" type="math/tex">n</script></span>-th roots of unity is linearly dependent for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&gt;</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-189-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.953em; display: inline-block;" role="math" id="MathJax-Span-2030" class="math"><span style="display: inline-block; position: relative; width: 2.26em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.15em, 2.812em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2031" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2032" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2033" class="mo">></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2034" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.094em; border-left: 0px solid; width: 0px; height: 1.035em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>1</mn></math></span></span><script id="MathJax-Element-189" type="math/tex">n > 1</script></span>, so multiplication is <em>not</em> the multiplication of the group ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mo fence="false" stretchy="false">&#x27E8;</mo><msub><mi>e</mi><mi>n</mi></msub><mo fence="false" stretchy="false">&#x27E9;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-190-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.141em; display: inline-block;" role="math" id="MathJax-Span-2035" class="math"><span style="display: inline-block; position: relative; width: 2.404em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.787em, 1002.33em, 3.001em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2036" class="mrow"><span id="MathJax-Span-2037" class="texatom"><span id="MathJax-Span-2038" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2039" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2040" class="mo">⟨</span><span id="MathJax-Span-2041" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2042" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2043" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2044" class="mo">⟩</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.339em; border-left: 0px solid; width: 0px; height: 1.329em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mo fence="false" stretchy="false">⟨</mo><msub><mi>e</mi><mi>n</mi></msub><mo fence="false" stretchy="false">⟩</mo></math></span></span><script id="MathJax-Element-190" type="math/tex">ℚ\langle e_n \rangle</script></span>; given a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-191-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-2045" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.72em, 2.853em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2046" class="mrow"><span id="MathJax-Span-2047" class="texatom"><span id="MathJax-Span-2048" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2049" class="mo">ℚ</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.147em; border-left: 0px solid; width: 0px; height: 1.088em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow></math></span></span><script id="MathJax-Element-191" type="math/tex">ℚ</script></span>-basis of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mo stretchy="false">(</mo><msub><mi>e</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-192-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.016em; display: inline-block;" role="math" id="MathJax-Span-2050" class="math"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.26em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2051" class="mrow"><span id="MathJax-Span-2052" class="texatom"><span id="MathJax-Span-2053" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2054" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2055" class="mo">(</span><span id="MathJax-Span-2056" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2057" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2058" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2059" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mo stretchy="false">(</mo><msub><mi>e</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-192" type="math/tex">ℚ(e_n)</script></span> the result of the multiplication (computed as multiplication of polynomials in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-193-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.141em; display: inline-block;" role="math" id="MathJax-Span-2060" class="math"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1000.86em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-2061" class="mrow"><span id="MathJax-Span-2062" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2063" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2064" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 0.902em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mi>n</mi></msub></math></span></span><script id="MathJax-Element-193" type="math/tex">e_n</script></span>, using <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>e</mi><mi>n</mi></msub><msup><mo stretchy="false">)</mo><mi>n</mi></msup><mo>=</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-194-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.891em; display: inline-block;" role="math" id="MathJax-Span-2065" class="math"><span style="display: inline-block; position: relative; width: 3.75em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.64em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2066" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2067" class="mo">(</span><span id="MathJax-Span-2068" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2069" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2070" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-2071" class="msubsup"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-2072" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2073" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2074" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2075" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>e</mi><mi>n</mi></msub><msup><mo stretchy="false">)</mo><mi>n</mi></msup><mo>=</mo><mn>1</mn></math></span></span><script id="MathJax-Element-194" type="math/tex">(e_n)^n = 1</script></span>) will be converted to the basis.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">E(5) * E(5)^2; ( E(5) + E(5)^4 ) * E(5)^2;</span>
-E(5)^3
-E(5)+E(5)^3
-<span class="GAPprompt">gap></span> <span class="GAPinput">( E(5) + E(5)^4 ) * E(5);</span>
--E(5)-E(5)^3-E(5)^4
-</pre></div>
-
-<p>An internally represented cyclotomic is always represented in the
-smallest cyclotomic field it is contained in. The internal coefficients
-list coincides with the external representation returned by <code class="func">ExtRepOfObj</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X785F2CAB805DE1BE"><span class="RefLink">18.1-12</span></a>).</p>
-
-<p>To avoid calculations becoming unintentionally very long, or
-consuming very large amounts of memory, there is a limit on the
-conductor of internally represented cyclotomics, by default set to one
-million. This can be raised (although not lowered) using <code class="func">SetCyclotomicsLimit</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7D3028777DE39709"><span class="RefLink">18.6-1</span></a>) and accessed using <code class="func">GetCyclotomicsLimit</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7D3028777DE39709"><span class="RefLink">18.6-1</span></a>). The maximum value of the limit is <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>28</mn></mrow></msup><mo>&#x2212;</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-195-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.828em; display: inline-block;" role="math" id="MathJax-Span-2076" class="math"><span style="display: inline-block; position: relative; width: 2.933em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.619em, 1002.83em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2077" class="mrow"><span id="MathJax-Span-2078" class="msubsup"><span style="display: inline-block; position: relative; width: 1.282em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.47em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-2079" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.393em; left: 0.5em;"><span id="MathJax-Span-2080" class="texatom"><span id="MathJax-Span-2081" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2082" class="mn">28</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2083" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2084" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.27em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>28</mn></mrow></msup><mo>−</mo><mn>1</mn></math></span></span><script id="MathJax-Element-195" type="math/tex">2^{28}-1</script></span> on <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-196-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-2085" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.94em, 2.802em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2086" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2087" class="mn">32</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.081em; border-left: 0px solid; width: 0px; height: 1.022em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn></math></span></span><script id="MathJax-Element-196" type="math/tex">32</script></span> bit systems, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>32</mn></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-197-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.703em; display: inline-block;" role="math" id="MathJax-Span-2088" class="math"><span style="display: inline-block; position: relative; width: 1.298em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.427em, 1001.3em, 2.596em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-2089" class="mrow"><span id="MathJax-Span-2090" class="msubsup"><span style="display: inline-block; position: relative; width: 1.282em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.47em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-2091" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.393em; left: 0.5em;"><span id="MathJax-Span-2092" class="texatom"><span id="MathJax-Span-2093" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2094" class="mn">32</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.27em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>32</mn></mrow></msup></math></span></span><script id="MathJax-Element-197" type="math/tex">2^{32}</script></span> on <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>64</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-198-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-2095" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.816em, 1000.93em, 2.802em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2096" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2097" class="mn">64</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.081em; border-left: 0px solid; width: 0px; height: 1.032em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>64</mn></math></span></span><script id="MathJax-Element-198" type="math/tex">64</script></span> bit systems. So the maximal cyclotomic field implemented in <strong class="pkg">GAP</strong> is not really the field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>b</mi></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-199-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.016em; display: inline-block;" role="math" id="MathJax-Span-2098" class="math"><span style="display: inline-block; position: relative; width: 1.538em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.422em, 1001.54em, 2.661em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-2099" class="mrow"><span id="MathJax-Span-2100" class="msubsup"><span style="display: inline-block; position: relative; width: 1.548em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-2101" class="texatom"><span id="MathJax-Span-2102" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2103" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.393em; left: 0.765em;"><span id="MathJax-Span-2104" class="texatom"><span id="MathJax-Span-2105" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2106" class="mi">a</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2107" class="mi">b</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.147em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mi>b</mi></mrow></msup></math></span></span><script id="MathJax-Element-199" type="math/tex">ℚ^{ab}</script></span>.</p>
-
-<p>It should be emphasized that one disadvantage of representing a cyclotomic in the smallest <em>cyclotomic</em>
- field (and not in the smallest field) is that arithmetic operations in a
- fixed small extension field of the rational number field are
-comparatively expensive. For example, take a prime integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-200-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-2108" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2109" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2110" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-200" type="math/tex">p</script></span> and suppose that we want to work with a matrix group over the field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mo stretchy="false">(</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mi>p</mi></mrow></msqrt><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-201-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.516em; display: inline-block;" role="math" id="MathJax-Span-2111" class="math"><span style="display: inline-block; position: relative; width: 2.692em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.64em, 3.133em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2112" class="mrow"><span id="MathJax-Span-2113" class="texatom"><span id="MathJax-Span-2114" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2115" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2116" class="mo">(</span><span id="MathJax-Span-2117" class="msqrt"><span style="display: inline-block; position: relative; width: 1.241em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.34em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-2118" class="mrow"><span id="MathJax-Span-2119" class="texatom"><span id="MathJax-Span-2120" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2121" class="mi">p</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1000.5em, 3.365em, -1000em); top: -3.78em; left: 0.737em;"><span style="font-family: STIXGeneral;">‾</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.657em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2122" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.51em; border-left: 0px solid; width: 0px; height: 1.451em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mo stretchy="false">(</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mi>p</mi></mrow></msqrt><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-201" type="math/tex">ℚ(\sqrt{{p}})</script></span>.
- Then each matrix entry could be described by two rational coefficients,
- whereas the representation in the smallest cyclotomic field requires <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#x2212;</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-202-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.766em; display: inline-block;" role="math" id="MathJax-Span-2123" class="math"><span style="display: inline-block; position: relative; width: 2.115em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.01em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2124" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2125" class="mi">p</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2126" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2127" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.27em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>−</mo><mn>1</mn></math></span></span><script id="MathJax-Element-202" type="math/tex">p-1</script></span> rational coefficients for each entry. So it is worth thinking about using elements in a field constructed with <code class="func">AlgebraicExtension</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap67_mj.html#X7CDA90537D2BAC8A"><span class="RefLink">67.1-1</span></a>) when natural embeddings of cyclotomic fields are not needed.</p>
-
-<p><a id="X7D3028777DE39709" name="X7D3028777DE39709"></a></p>
-
-<h5>18.6-1 SetCyclotomicsLimit</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SetCyclotomicsLimit</code>( <var class="Arg">newlimit</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ GetCyclotomicsLimit</code>( )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p><code class="func">GetCyclotomicsLimit</code> returns the current limit on conductors of internally represented cyclotomic numbers</p>
-
-<p><code class="func">SetCyclotomicsLimit</code> can be called to
-increase the limit on conductors of internally represented cyclotomic
-numbers. Note that computing in large cyclotomic fields using this
-representation can be both slow and memory-consuming, and that other
-approaches may be better for some problems. See <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8557FC2D7ACD6105"><span class="RefLink">18.6</span></a>.</p>
-
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href="http://www.gap-system.org/Manuals/doc/ref/chap64_mj.html">64</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap65_mj.html">65</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap66_mj.html">66</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap67_mj.html">67</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap68_mj.html">68</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap69_mj.html">69</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap70_mj.html">70</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html">71</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html">72</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html">73</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap74_mj.html">74</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap75_mj.html">75</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap76_mj.html">76</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap77_mj.html">77</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap78_mj.html">78</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap79_mj.html">79</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap80_mj.html">80</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap81_mj.html">81</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap82_mj.html">82</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap83_mj.html">83</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap84_mj.html">84</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap85_mj.html">85</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap86_mj.html">86</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap87_mj.html">87</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html">Bib</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chapInd_mj.html">Ind</a> </div>
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-<div class="ChapSects"><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80510B5880521FDC">60 <span class="Heading">Abelian Number Fields</span></a>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D4E43E5799753B5">60.1 <span class="Heading">Construction of Abelian Number Fields</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80D21D80850EFA4B">60.1-1 CyclotomicField</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80E5AD028143E11E">60.1-2 AbelianNumberField</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X82F53C65802FF551">60.1-3 GaussianRationals</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X81B5FE06781DB824">60.2 <span class="Heading">Operations for Abelian Number Fields</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7B0AB0FB7A4136C4">60.2-1 Factors</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X87D78F5E875F2E8A">60.2-2 IsNumberField</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D202D707D5708FA">60.2-3 IsAbelianNumberField</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X84CAE4627F0CD639">60.2-4 IsCyclotomicField</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X87E7313D8070B9CC">60.2-5 GaloisStabilizer</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE">60.3 <span class="Heading">Integral Bases of Abelian Number Fields</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7F52BEA0862E06F2">60.3-1 ZumbroichBase</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X87DB9C2C858B722A">60.3-2 LenstraBase</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7E4AB4B17C7BA10C">60.4 <span class="Heading">Galois Groups of Abelian Number Fields</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7B55A90582E818F3">60.4-1 GaloisGroup</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X8643D4B47A827D9D">60.4-2 ANFAutomorphism</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X85E9E90D7FE877CC">60.5 <span class="Heading">Gaussians</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80BD5EAB879F096E">60.5-1 GaussianIntegers</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7BFD33D27BFB7C5A">60.5-2 IsGaussianIntegers</a></span>
-</div></div>
-</div>
-
-<h3>60 <span class="Heading">Abelian Number Fields</span></h3>
-
-<p>An <em>abelian number field</em> is a field in characteristic zero
-that is a finite dimensional normal extension of its prime field such
-that the Galois group is abelian. In <strong class="pkg">GAP</strong>, one implementation of abelian number fields is given by fields of cyclotomic numbers (see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>). Note that abelian number fields can also be constructed with the more general <code class="func">AlgebraicExtension</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap67_mj.html#X7CDA90537D2BAC8A"><span class="RefLink">67.1-1</span></a>), a discussion of advantages and disadvantages can be found in <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8557FC2D7ACD6105"><span class="RefLink">18.6</span></a>. The functions described in this chapter have been developed for fields whose elements are in the filter <code class="func">IsCyclotomic</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X841C425281A6F775"><span class="RefLink">18.1-3</span></a>), they may or may not work well for abelian number fields consisting of other kinds of elements.</p>
-
-<p>Throughout this chapter, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-1-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-1" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-2" class="mrow"><span id="MathJax-Span-3" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-4" class="texatom"><span id="MathJax-Span-5" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-6" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-7" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-1" type="math/tex">â„š_n</script></span> will denote the cyclotomic field generated by the field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-2-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-8" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.72em, 2.853em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-9" class="mrow"><span id="MathJax-Span-10" class="texatom"><span id="MathJax-Span-11" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-12" class="mo">â„š</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.147em; border-left: 0px solid; width: 0px; height: 1.088em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow></math></span></span><script id="MathJax-Element-2" type="math/tex">â„š</script></span> of rationals together with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-3-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-13" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-14" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-15" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-3" type="math/tex">n</script></span>-th roots of unity.</p>
-
-<p>In <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D4E43E5799753B5"><span class="RefLink">60.1</span></a>, constructors for abelian number fields are described, <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X81B5FE06781DB824"><span class="RefLink">60.2</span></a> introduces operations for abelian number fields, <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a> deals with the vector space structure of abelian number fields, and <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7E4AB4B17C7BA10C"><span class="RefLink">60.4</span></a> describes field automorphisms of abelian number fields,</p>
-
-<p><a id="X7D4E43E5799753B5" name="X7D4E43E5799753B5"></a></p>
-
-<h4>60.1 <span class="Heading">Construction of Abelian Number Fields</span></h4>
-
-<p>Besides the usual construction using <code class="func">Field</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html#X871AA7D58263E9AC"><span class="RefLink">58.1-3</span></a>) or <code class="func">DefaultField</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7FE3D5637B5485D0"><span class="RefLink">18.1-16</span></a>) (see <code class="func">DefaultField</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7FE3D5637B5485D0"><span class="RefLink">18.1-16</span></a>)), abelian number fields consisting of cyclotomics can be created with <code class="func">CyclotomicField</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) and <code class="func">AbelianNumberField</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>).</p>
-
-<p><a id="X80D21D80850EFA4B" name="X80D21D80850EFA4B"></a></p>
-
-<h5>60.1-1 CyclotomicField</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CyclotomicField</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CyclotomicField</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CF</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CF</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>The first version creates the <var class="Arg">n</var>-th cyclotomic field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-4-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-16" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-17" class="mrow"><span id="MathJax-Span-18" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-19" class="texatom"><span id="MathJax-Span-20" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-21" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-22" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-4" type="math/tex">â„š_n</script></span>. The second version creates the smallest cyclotomic field containing the elements in the list <var class="Arg">gens</var>. In both cases the field can be generated as an extension of a designated subfield <var class="Arg">subfield</var> (cf. <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p>
-
-<p><code class="func">CyclotomicField</code> can be abbreviated to <code class="func">CF</code>, this form is used also when <strong class="pkg">GAP</strong> prints cyclotomic fields.</p>
-
-<p>Fields constructed with the one argument version of <code class="func">CF</code> are stored in the global list <code class="code">CYCLOTOMIC_FIELDS</code>, so repeated calls of <code class="func">CF</code> just fetch these field objects after they have been created once.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">CyclotomicField( 5 ); CyclotomicField( [ Sqrt(3) ] );</span>
-CF(5)
-CF(12)
-<span class="GAPprompt">gap></span> <span class="GAPinput">CF( CF(3), 12 ); CF( CF(4), [ Sqrt(7) ] );</span>
-AsField( CF(3), CF(12) )
-AsField( GaussianRationals, CF(28) )
-</pre></div>
-
-<p><a id="X80E5AD028143E11E" name="X80E5AD028143E11E"></a></p>
-
-<h5>60.1-2 AbelianNumberField</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ AbelianNumberField</code>( <var class="Arg">n</var>, <var class="Arg">stab</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ NF</code>( <var class="Arg">n</var>, <var class="Arg">stab</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>For a positive integer <var class="Arg">n</var> and a list <var class="Arg">stab</var> of prime residues modulo <var class="Arg">n</var>, <code class="func">AbelianNumberField</code> returns the fixed field of the group described by <var class="Arg">stab</var> (cf. <code class="func">GaloisStabilizer</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X87E7313D8070B9CC"><span class="RefLink">60.2-5</span></a>)), in the <var class="Arg">n</var>-th cyclotomic field. <code class="func">AbelianNumberField</code> is mainly thought for internal use and for printing fields in a standard way; <code class="func">Field</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html#X871AA7D58263E9AC"><span class="RefLink">58.1-3</span></a>) (cf. also <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X81B5FE06781DB824"><span class="RefLink">60.2</span></a>) is probably more suitable if one knows generators of the field in question.</p>
-
-<p><code class="func">AbelianNumberField</code> can be abbreviated to <code class="func">NF</code>, this form is used also when <strong class="pkg">GAP</strong> prints abelian number fields.</p>
-
-<p>Fields constructed with <code class="func">NF</code> are stored in the global list <code class="code">ABELIAN_NUMBER_FIELDS</code>, so repeated calls of <code class="func">NF</code> just fetch these field objects after they have been created once.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">NF( 7, [ 1 ] );</span>
-CF(7)
-<span class="GAPprompt">gap></span> <span class="GAPinput">f:= NF( 7, [ 1, 2 ] ); Sqrt(-7); Sqrt(-7) in f;</span>
-NF(7,[ 1, 2, 4 ])
-E(7)+E(7)^2-E(7)^3+E(7)^4-E(7)^5-E(7)^6
-true
-</pre></div>
-
-<p><a id="X82F53C65802FF551" name="X82F53C65802FF551"></a></p>
-
-<h5>60.1-3 GaussianRationals</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ GaussianRationals</code></td><td class="tdright">( global variable )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsGaussianRationals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<p><code class="func">GaussianRationals</code> is the field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mn>4</mn></msub><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mo stretchy="false">(</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msqrt><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-5-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.641em; display: inline-block;" role="math" id="MathJax-Span-23" class="math"><span style="display: inline-block; position: relative; width: 5.865em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.67em, 1005.82em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-24" class="mrow"><span id="MathJax-Span-25" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-26" class="texatom"><span id="MathJax-Span-27" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-28" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-29" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-30" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-31" class="texatom"><span id="MathJax-Span-32" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-33" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-34" class="mo">(</span><span id="MathJax-Span-35" class="msqrt"><span style="display: inline-block; position: relative; width: 1.947em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1001.08em, 4.135em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-36" class="mrow"><span id="MathJax-Span-37" class="texatom"><span id="MathJax-Span-38" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-39" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-40" class="mn">1</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1001.21em, 3.365em, -1000em); top: -4em; left: 0.737em;"><span style="display: inline-block; position: relative; width: 1.21em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0.71em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.99em; left: 0.343em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.877em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-41" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.434em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mn>4</mn></msub><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mo stretchy="false">(</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msqrt><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-5" type="math/tex">ℚ_4 = ℚ(\sqrt{{-1}})</script></span> of Gaussian rationals, as a set of cyclotomic numbers, see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a> for basic operations. This field can also be obtained as <code class="code">CF(4)</code> (see <code class="func">CyclotomicField</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>)).</p>
-
-<p>The filter <code class="func">IsGaussianRationals</code> returns <code class="keyw">true</code> for the <strong class="pkg">GAP</strong> object <code class="func">GaussianRationals</code>, and <code class="keyw">false</code> for all other <strong class="pkg">GAP</strong> objects.</p>
-
-<p>(For details about the field of rationals, see Chapter <code class="func">Rationals</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap17_mj.html#X7B6029D18570C08A"><span class="RefLink">17.1-1</span></a>).)</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">CF(4) = GaussianRationals;</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">Sqrt(-1) in GaussianRationals;</span>
-true
-</pre></div>
-
-<p><a id="X81B5FE06781DB824" name="X81B5FE06781DB824"></a></p>
-
-<h4>60.2 <span class="Heading">Operations for Abelian Number Fields</span></h4>
-
-<p>For operations for elements of abelian number fields, e.g., <code class="func">Conductor</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>) or <code class="func">ComplexConjugate</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7BE001A0811CD599"><span class="RefLink">18.5-2</span></a>), see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>.</p>
-
-<p><a id="X7B0AB0FB7A4136C4" name="X7B0AB0FB7A4136C4"></a></p>
-
-<h5>60.2-1 Factors</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Factors</code>( <var class="Arg">F</var> )</td><td class="tdright">( method )</td></tr></tbody></table></div>
-<p>Factoring of polynomials over abelian number fields consisting of
-cyclotomics works in principle but is not very efficient if the degree
-of the field extension is large.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">x:= Indeterminate( CF(5) );</span>
-x_1
-<span class="GAPprompt">gap></span> <span class="GAPinput">Factors( PolynomialRing( Rationals ), x^5-1 );</span>
-[ x_1-1, x_1^4+x_1^3+x_1^2+x_1+1 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Factors( PolynomialRing( CF(5) ), x^5-1 );</span>
-[ x_1-1, x_1+(-E(5)), x_1+(-E(5)^2), x_1+(-E(5)^3), x_1+(-E(5)^4) ]
-</pre></div>
-
-<p><a id="X87D78F5E875F2E8A" name="X87D78F5E875F2E8A"></a></p>
-
-<h5>60.2-2 IsNumberField</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsNumberField</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<p>returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a finite dimensional extension of a prime field in characteristic zero, and <code class="keyw">false</code> otherwise.</p>
-
-<p><a id="X7D202D707D5708FA" name="X7D202D707D5708FA"></a></p>
-
-<h5>60.2-3 IsAbelianNumberField</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsAbelianNumberField</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<p>returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a number field (see <code class="func">IsNumberField</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X87D78F5E875F2E8A"><span class="RefLink">60.2-2</span></a>)) that is a Galois extension of the prime field, with abelian Galois group (see <code class="func">GaloisGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html#X80CAA5BA82F09ED2"><span class="RefLink">58.3-1</span></a>)).</p>
-
-<p><a id="X84CAE4627F0CD639" name="X84CAE4627F0CD639"></a></p>
-
-<h5>60.2-4 IsCyclotomicField</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsCyclotomicField</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<p>returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a <em>cyclotomic field</em>, i.e., an abelian number field (see <code class="func">IsAbelianNumberField</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D202D707D5708FA"><span class="RefLink">60.2-3</span></a>)) that can be generated by roots of unity.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">IsNumberField( CF(9) ); IsAbelianNumberField( Field( [ ER(3) ] ) );</span>
-true
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsNumberField( GF(2) );</span>
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( CF(9) );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( Field( [ Sqrt(-3) ] ) );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( Field( [ Sqrt(3) ] ) );</span>
-false
-</pre></div>
-
-<p><a id="X87E7313D8070B9CC" name="X87E7313D8070B9CC"></a></p>
-
-<h5>60.2-5 GaloisStabilizer</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ GaloisStabilizer</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">F</var> be an abelian number field (see <code class="func">IsAbelianNumberField</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D202D707D5708FA"><span class="RefLink">60.2-3</span></a>)) with conductor <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-6-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-42" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-43" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-44" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-6" type="math/tex">n</script></span>, say. (This means that the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-7-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-45" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-46" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-47" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-7" type="math/tex">n</script></span>-th cyclotomic field is the smallest cyclotomic field containing <var class="Arg">F</var>, see <code class="func">Conductor</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>).) <code class="func">GaloisStabilizer</code> returns the set of all those integers <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-8-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-48" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.43em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-49" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-50" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script id="MathJax-Element-8" type="math/tex">k</script></span> in the range <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>1</mn><mo>.</mo><mo>.</mo><mi>n</mi><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-9-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.266em; display: inline-block;" role="math" id="MathJax-Span-51" class="math"><span style="display: inline-block; position: relative; width: 2.5em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.41em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-52" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-53" class="mo">[</span><span style="font-family: STIXGeneral;" id="MathJax-Span-54" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-55" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-56" class="mo">.</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-57" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-58" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>1</mn><mo>.</mo><mo>.</mo><mi>n</mi><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-9" type="math/tex">[ 1 .. n ]</script></span> such that the field automorphism induced by raising <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-10-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-59" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-60" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-61" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-10" type="math/tex">n</script></span>-th roots of unity to the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-11-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-62" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.43em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-63" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-64" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script id="MathJax-Element-11" type="math/tex">k</script></span>-th power acts trivially on <var class="Arg">F</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">r5:= Sqrt(5);</span>
-E(5)-E(5)^2-E(5)^3+E(5)^4
-<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( r5, 4 ) = r5; GaloisCyc( r5, 2 ) = r5;</span>
-true
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisStabilizer( Field( [ r5 ] ) );</span>
-[ 1, 4 ]
-</pre></div>
-
-<p><a id="X7D2421AC8491D2BE" name="X7D2421AC8491D2BE"></a></p>
-
-<h4>60.3 <span class="Heading">Integral Bases of Abelian Number Fields</span></h4>
-
-<p>Each abelian number field is naturally a vector space over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-12-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-65" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.72em, 2.853em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-66" class="mrow"><span id="MathJax-Span-67" class="texatom"><span id="MathJax-Span-68" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-69" class="mo">â„š</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.147em; border-left: 0px solid; width: 0px; height: 1.088em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow></math></span></span><script id="MathJax-Element-12" type="math/tex">â„š</script></span>. Moreover, if the abelian number field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-13-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-70" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.67em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-71" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-72" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math></span></span><script id="MathJax-Element-13" type="math/tex">F</script></span> contains the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-14-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-73" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-74" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-75" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-14" type="math/tex">n</script></span>-th cyclotomic field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-15-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-76" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-77" class="mrow"><span id="MathJax-Span-78" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-79" class="texatom"><span id="MathJax-Span-80" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-81" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-82" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-15" type="math/tex">â„š_n</script></span> then <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-16-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-83" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.67em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-84" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-85" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math></span></span><script id="MathJax-Element-16" type="math/tex">F</script></span> is a vector space over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-17-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-86" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-87" class="mrow"><span id="MathJax-Span-88" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-89" class="texatom"><span id="MathJax-Span-90" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-91" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-92" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-17" type="math/tex">â„š_n</script></span>. In <strong class="pkg">GAP</strong>, each field object represents a vector space object over a certain subfield <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-18-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-93" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.833em, 1000.48em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-94" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-95" class="mi">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.016em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script id="MathJax-Element-18" type="math/tex">S</script></span>, which depends on the way <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-19-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-96" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.67em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-97" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-98" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math></span></span><script id="MathJax-Element-19" type="math/tex">F</script></span> was constructed. The subfield <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-20-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-99" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.833em, 1000.48em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-100" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-101" class="mi">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.016em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script id="MathJax-Element-20" type="math/tex">S</script></span> can be accessed as the value of the attribute <code class="func">LeftActingDomain</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap57_mj.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>).</p>
-
-<p>The return values of <code class="func">NF</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>) and of the one argument versions of <code class="func">CF</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) represent vector spaces over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-21-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-102" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.72em, 2.853em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-103" class="mrow"><span id="MathJax-Span-104" class="texatom"><span id="MathJax-Span-105" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-106" class="mo">â„š</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.147em; border-left: 0px solid; width: 0px; height: 1.088em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow></math></span></span><script id="MathJax-Element-21" type="math/tex">â„š</script></span>, and the return values of the two argument version of <code class="func">CF</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) represent vector spaces over the field that is given as the first argument. For an abelian number field <var class="Arg">F</var> and a subfield <var class="Arg">S</var> of <var class="Arg">F</var>, a <strong class="pkg">GAP</strong> object representing <var class="Arg">F</var> as a vector space over <var class="Arg">S</var> can be constructed using <code class="func">AsField</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html#X7C193B7D7AFB29BE"><span class="RefLink">58.1-9</span></a>).</p>
-
-<p>Let <var class="Arg">F</var> be the cyclotomic field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-22-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-107" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-108" class="mrow"><span id="MathJax-Span-109" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-110" class="texatom"><span id="MathJax-Span-111" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-112" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-113" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-22" type="math/tex">â„š_n</script></span>, represented as a vector space over the subfield <var class="Arg">S</var>. If <var class="Arg">S</var> is the cyclotomic field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>m</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-23-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.766em; display: inline-block;" role="math" id="MathJax-Span-114" class="math"><span style="display: inline-block; position: relative; width: 1.346em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.35em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-115" class="mrow"><span id="MathJax-Span-116" class="msubsup"><span style="display: inline-block; position: relative; width: 1.351em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-117" class="texatom"><span id="MathJax-Span-118" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-119" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-120" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>m</mi></msub></math></span></span><script id="MathJax-Element-23" type="math/tex">â„š_m</script></span>, with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-24-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-121" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.7em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-122" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-123" class="mi">m</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script id="MathJax-Element-24" type="math/tex">m</script></span> a divisor of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-25-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-124" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-125" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-126" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-25" type="math/tex">n</script></span>, then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns the Zumbroich basis of <var class="Arg">F</var> relative to <var class="Arg">S</var>, which consists of the roots of unity <code class="code">E(<var class="Arg">n</var>)</code>^<var class="Arg">i</var> where <var class="Arg">i</var> is an element of the list <code class="code">ZumbroichBase( <var class="Arg">n</var>, <var class="Arg">m</var> )</code> (see <code class="func">ZumbroichBase</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7F52BEA0862E06F2"><span class="RefLink">60.3-1</span></a>)). If <var class="Arg">S</var> is an abelian number field that is not a cyclotomic field then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns a normal <var class="Arg">S</var>-basis of <var class="Arg">F</var>, i.e., a basis that is closed under the field automorphisms of <var class="Arg">F</var>.</p>
-
-<p>Let <var class="Arg">F</var> be the abelian number field <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stab</var> )</code>, with conductor <var class="Arg">n</var>, that is itself not a cyclotomic field, represented as a vector space over the subfield <var class="Arg">S</var>. If <var class="Arg">S</var> is the cyclotomic field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>m</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-26-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.766em; display: inline-block;" role="math" id="MathJax-Span-127" class="math"><span style="display: inline-block; position: relative; width: 1.346em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.35em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-128" class="mrow"><span id="MathJax-Span-129" class="msubsup"><span style="display: inline-block; position: relative; width: 1.351em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-130" class="texatom"><span id="MathJax-Span-131" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-132" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-133" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>m</mi></msub></math></span></span><script id="MathJax-Element-26" type="math/tex">â„š_m</script></span>, with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-27-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-134" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.7em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-135" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-136" class="mi">m</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script id="MathJax-Element-27" type="math/tex">m</script></span> a divisor of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-28-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-137" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-138" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-139" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-28" type="math/tex">n</script></span>, then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns the Lenstra basis of <var class="Arg">F</var> relative to <var class="Arg">S</var> that consists of the sums of roots of unity described by <code class="code">LenstraBase( <var class="Arg">n</var>, <var class="Arg">stab</var>, <var class="Arg">stab</var>, <var class="Arg">m</var> )</code> (see <code class="func">LenstraBase</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X87DB9C2C858B722A"><span class="RefLink">60.3-2</span></a>)). If <var class="Arg">S</var> is an abelian number field that is not a cyclotomic field then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns a normal <var class="Arg">S</var>-basis of <var class="Arg">F</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(8);; # a cycl. field over the rationals</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
-[ 1, E(8), E(4), E(8)^3 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
-[ 0, 1, 0, 1 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AsField( CF(4), CF(8) );; # a cycl. field over a cycl. field</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
-[ 1, E(8) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
-[ 0, 1+E(4) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AsField( Field( [ Sqrt(-2) ] ), CF(8) );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput"># a cycl. field over a non-cycl. field</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
-[ 1/2+1/2*E(8)-1/2*E(8)^2-1/2*E(8)^3,
- 1/2-1/2*E(8)+1/2*E(8)^2+1/2*E(8)^3 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
-[ E(8)+E(8)^3, E(8)+E(8)^3 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">f:= Field( [ Sqrt(-2) ] ); # a non-cycl. field over the rationals</span>
-NF(8,[ 1, 3 ])
-<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
-[ 1, E(8)+E(8)^3 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
-[ 0, 1 ]
-</pre></div>
-
-<p><a id="X7F52BEA0862E06F2" name="X7F52BEA0862E06F2"></a></p>
-
-<h5>60.3-1 ZumbroichBase</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ZumbroichBase</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">n</var> and <var class="Arg">m</var> be positive integers, such that <var class="Arg">m</var> divides <var class="Arg">n</var>. <code class="func">ZumbroichBase</code> returns the set of exponents <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-29-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-140" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-141" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-142" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-29" type="math/tex">i</script></span> for which <code class="code">E(<var class="Arg">n</var>)^</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-30-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-143" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-144" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-145" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-30" type="math/tex">i</script></span> belongs to the (generalized) Zumbroich basis of the cyclotomic field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-31-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-146" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-147" class="mrow"><span id="MathJax-Span-148" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-149" class="texatom"><span id="MathJax-Span-150" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-151" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-152" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-31" type="math/tex">â„š_n</script></span>, viewed as a vector space over <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>m</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-32-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.766em; display: inline-block;" role="math" id="MathJax-Span-153" class="math"><span style="display: inline-block; position: relative; width: 1.346em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.35em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-154" class="mrow"><span id="MathJax-Span-155" class="msubsup"><span style="display: inline-block; position: relative; width: 1.351em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-156" class="texatom"><span id="MathJax-Span-157" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-158" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-159" class="mi">m</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>m</mi></msub></math></span></span><script id="MathJax-Element-32" type="math/tex">â„š_m</script></span>.</p>
-
-<p>This basis is defined as follows. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-33-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.828em; display: inline-block;" role="math" id="MathJax-Span-160" class="math"><span style="display: inline-block; position: relative; width: 0.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.62em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-161" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-162" class="mi">P</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></span></span><script id="MathJax-Element-33" type="math/tex">P</script></span> denote the set of prime divisors of <var class="Arg">n</var>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">n</mtext></mrow><mo>=</mo><munder><mo>&#x220F;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>p</mi><mo>&#x2208;</mo><mi>P</mi></mrow></mrow></munder><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>&#x03BD;</mi><mi>p</mi></msub></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-34-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.203em; display: inline-block;" role="math" id="MathJax-Span-163" class="math"><span style="display: inline-block; position: relative; width: 5.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.739em, 1005.53em, 3.23em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-164" class="mrow"><span id="MathJax-Span-165" class="texatom"><span id="MathJax-Span-166" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-167" class="mtext">n</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-168" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-169" class="munderover"><span style="display: inline-block; position: relative; width: 2.348em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.95em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-170" class="mo">âˆ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.694em; left: 1em;"><span id="MathJax-Span-171" class="texatom"><span id="MathJax-Span-172" class="mrow"><span id="MathJax-Span-173" class="texatom"><span id="MathJax-Span-174" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-175" class="mi">p</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-176" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-177" class="mi">P</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-178" class="msubsup"><span style="display: inline-block; position: relative; width: 1.216em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.34em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-179" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.504em;"><span id="MathJax-Span-180" class="texatom"><span id="MathJax-Span-181" class="mrow"><span id="MathJax-Span-182" class="texatom"><span id="MathJax-Span-183" class="mrow"><span id="MathJax-Span-184" class="msubsup"><span style="display: inline-block; position: relative; width: 0.637em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.33em, 4.147em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-185" class="mi">ν</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.332em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-186" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.636em; border-left: 0px solid; width: 0px; height: 1.688em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">n</mtext></mrow><mo>=</mo><munder><mo>âˆ</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>p</mi><mo>∈</mo><mi>P</mi></mrow></mrow></munder><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>ν</mi><mi>p</mi></msub></mrow></mrow></msup></math></span></span><script id="MathJax-Element-34" type="math/tex">\textit{n} = \prod_{{p \in P}} p^{{\nu_p}}</script></span>, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">m</mtext></mrow><mo>=</mo><munder><mo>&#x220F;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>p</mi><mo>&#x2208;</mo><mi>P</mi></mrow></mrow></munder><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>&#x03BC;</mi><mi>p</mi></msub></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-35-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.578em; display: inline-block;" role="math" id="MathJax-Span-187" class="math"><span style="display: inline-block; position: relative; width: 5.817em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.739em, 1005.82em, 3.23em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-188" class="mrow"><span id="MathJax-Span-189" class="texatom"><span id="MathJax-Span-190" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-191" class="mtext">m</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-192" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-193" class="munderover"><span style="display: inline-block; position: relative; width: 2.348em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.95em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-194" class="mo">âˆ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.694em; left: 1em;"><span id="MathJax-Span-195" class="texatom"><span id="MathJax-Span-196" class="mrow"><span id="MathJax-Span-197" class="texatom"><span id="MathJax-Span-198" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-199" class="mi">p</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-200" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-201" class="mi">P</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-202" class="msubsup"><span style="display: inline-block; position: relative; width: 1.256em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.34em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-203" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.37em; left: 0.504em;"><span id="MathJax-Span-204" class="texatom"><span id="MathJax-Span-205" class="mrow"><span id="MathJax-Span-206" class="texatom"><span id="MathJax-Span-207" class="mrow"><span id="MathJax-Span-208" class="msubsup"><span style="display: inline-block; position: relative; width: 0.677em; height: 0px;"><span style="position: absolute; clip: rect(3.544em, 1000.34em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-209" class="mi">μ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.82em; left: 0.372em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-210" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.636em; border-left: 0px solid; width: 0px; height: 1.688em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">m</mtext></mrow><mo>=</mo><munder><mo>âˆ</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>p</mi><mo>∈</mo><mi>P</mi></mrow></mrow></munder><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>μ</mi><mi>p</mi></msub></mrow></mrow></msup></math></span></span><script id="MathJax-Element-35" type="math/tex">\textit{m} = \prod_{{p \in P}} p^{{\mu_p}}</script></span> with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03BC;</mi><mi>p</mi></msub><mo>&#x2264;</mo><msub><mi>&#x03BD;</mi><mi>p</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-36-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.141em; display: inline-block;" role="math" id="MathJax-Span-211" class="math"><span style="display: inline-block; position: relative; width: 3.173em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.891em, 1003.17em, 3.083em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-212" class="mrow"><span id="MathJax-Span-213" class="msubsup"><span style="display: inline-block; position: relative; width: 0.957em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.48em, 4.34em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-214" class="mi">μ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-215" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-216" class="mo">≤</span><span style="padding-left: 0.313em;" id="MathJax-Span-217" class="msubsup"><span style="display: inline-block; position: relative; width: 0.901em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.46em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-218" class="mi">ν</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.47em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-219" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.446em; border-left: 0px solid; width: 0px; height: 1.3em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>p</mi></msub><mo>≤</mo><msub><mi>ν</mi><mi>p</mi></msub></math></span></span><script id="MathJax-Element-36" type="math/tex">\mu_p \leq \nu_p</script></span>. Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mi>l</mi></msub><mo>=</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-37-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.203em; display: inline-block;" role="math" id="MathJax-Span-220" class="math"><span style="display: inline-block; position: relative; width: 1.683em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1001.64em, 2.946em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-221" class="mrow"><span id="MathJax-Span-222" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-223" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-224" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-225" class="mo">=</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 0.903em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mi>l</mi></msub><mo>=</mo></math></span></span><script id="MathJax-Element-37" type="math/tex">e_l =</script></span> <code class="code">E</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-38-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-226" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.91em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-227" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-228" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-229" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-230" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-38" type="math/tex">(l)</script></span> for any positive integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-39-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-231" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.29em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-232" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-233" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math></span></span><script id="MathJax-Element-39" type="math/tex">l</script></span>, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mn>1</mn></msub></mrow></mrow><mi>j</mi></msubsup><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>&#x2208;</mo><mi>J</mi></mrow></mrow></msub><mo>&#x2297;</mo><mo fence="false" stretchy="false">{</mo><msubsup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mn>2</mn></msub></mrow></mrow><mi>k</mi></msubsup><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>&#x2208;</mo><mi>K</mi></mrow></mrow></msub><mo>=</mo><mo fence="false" stretchy="false">{</mo><msubsup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mn>1</mn></msub></mrow></mrow><mi>j</mi></msubsup><mo>&#x22C5;</mo><msubsup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mn>2</mn></msub></mrow></mrow><mi>k</mi></msubsup><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>&#x2208;</mo><mi>J</mi><mo>,</mo><mi>k</mi><mo>&#x2208;</mo><mi>K</mi></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-40-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 21.016em; display: inline-block;" role="math" id="MathJax-Span-234" class="math"><span style="display: inline-block; position: relative; width: 16.154em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.548em, 1016.15em, 3.085em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-235" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-236" class="mo">{</span><span id="MathJax-Span-237" class="msubsup"><span style="display: inline-block; position: relative; width: 1.176em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-238" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.385em, 1000.27em, 4.281em, -1000em); top: -4.481em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-239" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.73em, 4.241em, -1000em); top: -3.843em; left: 0.444em;"><span id="MathJax-Span-240" class="texatom"><span id="MathJax-Span-241" class="mrow"><span id="MathJax-Span-242" class="texatom"><span id="MathJax-Span-243" class="mrow"><span id="MathJax-Span-244" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-245" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-246" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-247" class="msubsup"><span style="display: inline-block; position: relative; width: 1.584em; height: 0px;"><span style="position: absolute; clip: rect(3.166em, 1000.38em, 4.316em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-248" class="mo">}</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.48em;"><span id="MathJax-Span-249" class="texatom"><span id="MathJax-Span-250" class="mrow"><span id="MathJax-Span-251" class="texatom"><span id="MathJax-Span-252" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-253" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-254" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-255" class="mi">J<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.033em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-256" class="mo">⊗</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-257" class="mo">{</span><span id="MathJax-Span-258" class="msubsup"><span style="display: inline-block; position: relative; width: 1.176em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-259" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.4em, 4.142em, -1000em); top: -4.343em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-260" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.73em, 4.241em, -1000em); top: -3.843em; left: 0.444em;"><span id="MathJax-Span-261" class="texatom"><span id="MathJax-Span-262" class="mrow"><span id="MathJax-Span-263" class="texatom"><span id="MathJax-Span-264" class="mrow"><span id="MathJax-Span-265" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-266" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-267" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-268" class="msubsup"><span style="display: inline-block; position: relative; width: 1.876em; height: 0px;"><span style="position: absolute; clip: rect(3.166em, 1000.38em, 4.316em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-269" class="mo">}</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.48em;"><span id="MathJax-Span-270" class="texatom"><span id="MathJax-Span-271" class="mrow"><span id="MathJax-Span-272" class="texatom"><span id="MathJax-Span-273" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-274" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-275" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-276" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-277" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-278" class="mo">{</span><span id="MathJax-Span-279" class="msubsup"><span style="display: inline-block; position: relative; width: 1.176em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-280" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.385em, 1000.27em, 4.281em, -1000em); top: -4.481em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-281" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.73em, 4.241em, -1000em); top: -3.843em; left: 0.444em;"><span id="MathJax-Span-282" class="texatom"><span id="MathJax-Span-283" class="mrow"><span id="MathJax-Span-284" class="texatom"><span id="MathJax-Span-285" class="mrow"><span id="MathJax-Span-286" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-287" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-288" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-289" class="mo">â‹…</span><span style="padding-left: 0.25em;" id="MathJax-Span-290" class="msubsup"><span style="display: inline-block; position: relative; width: 1.176em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-291" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.4em, 4.142em, -1000em); top: -4.343em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-292" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.534em, 1000.73em, 4.241em, -1000em); top: -3.843em; left: 0.444em;"><span id="MathJax-Span-293" class="texatom"><span id="MathJax-Span-294" class="mrow"><span id="MathJax-Span-295" class="texatom"><span id="MathJax-Span-296" class="mrow"><span id="MathJax-Span-297" class="msubsup"><span style="display: inline-block; position: relative; width: 0.657em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.34em, 4.141em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-298" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-299" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-300" class="msubsup"><span style="display: inline-block; position: relative; width: 3.081em; height: 0px;"><span style="position: absolute; clip: rect(3.166em, 1000.38em, 4.316em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-301" class="mo">}</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.48em;"><span id="MathJax-Span-302" class="texatom"><span id="MathJax-Span-303" class="mrow"><span id="MathJax-Span-304" class="texatom"><span id="MathJax-Span-305" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-306" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-307" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-308" class="mi">J<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.033em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-309" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-310" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-311" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-312" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.448em; border-left: 0px solid; width: 0px; height: 1.748em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msubsup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mn>1</mn></msub></mrow></mrow><mi>j</mi></msubsup><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></mrow></msub><mo>⊗</mo><mo fence="false" stretchy="false">{</mo><msubsup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mn>2</mn></msub></mrow></mrow><mi>k</mi></msubsup><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>∈</mo><mi>K</mi></mrow></mrow></msub><mo>=</mo><mo fence="false" stretchy="false">{</mo><msubsup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mn>1</mn></msub></mrow></mrow><mi>j</mi></msubsup><mo>â‹…</mo><msubsup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mn>2</mn></msub></mrow></mrow><mi>k</mi></msubsup><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>∈</mo><mi>J</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi>K</mi></mrow></mrow></msub></math></span></span><script id="MathJax-Element-40" type="math/tex">\{ e_{{n_1}}^j \}_{{j \in J}} \otimes \{ e_{{n_2}}^k \}_{{k \in K}} = \{ e_{{n_1}}^j \cdot e_{{n_2}}^k \}_{{j \in J, k \in K}}</script></span>.</p>
-
-<p>Then the basis is</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"></span><div style="text-align: center;" class="MathJax_Display"><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>,</mo><mi>m</mi></mrow></mrow></msub><mo>=</mo><munder><mo>&#x2A02;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>p</mi><mo>&#x2208;</mo><mi>P</mi></mrow></mrow></munder><munderover><mo>&#x2A02;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><msub><mi>&#x03BC;</mi><mi>p</mi></msub></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>&#x03BD;</mi><mi>p</mi></msub><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></munderover><mo fence="false" stretchy="false">{</mo><msubsup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mrow></msup></mrow></mrow><mi>j</mi></msubsup><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>&#x2208;</mo><msub><mi>J</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>,</mo><mi>p</mi></mrow></mrow></msub></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-41-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 14.016em; display: inline-block;" role="math" id="MathJax-Span-313" class="math"><span style="display: inline-block; position: relative; width: 10.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(0.613em, 1010.77em, 4.311em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-314" class="mrow"><span id="MathJax-Span-315" class="msubsup"><span style="display: inline-block; position: relative; width: 1.727em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.59em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-316" class="mi">B</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span id="MathJax-Span-317" class="texatom"><span id="MathJax-Span-318" class="mrow"><span id="MathJax-Span-319" class="texatom"><span id="MathJax-Span-320" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-321" class="mi">n</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-322" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-323" class="mi">m</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-324" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-325" class="munderover"><span style="display: inline-block; position: relative; width: 1.555em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.5em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-326" class="mo">⨂</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1001.27em, 4.38em, -1000em); top: -2.84em; left: 0.141em;"><span id="MathJax-Span-327" class="texatom"><span id="MathJax-Span-328" class="mrow"><span id="MathJax-Span-329" class="texatom"><span id="MathJax-Span-330" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-331" class="mi">p</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-332" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-333" class="mi">P</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-334" class="munderover"><span style="display: inline-block; position: relative; width: 1.555em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.5em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-335" class="mo">⨂</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1001.49em, 4.507em, -1000em); top: -2.84em; left: 0.034em;"><span id="MathJax-Span-336" class="texatom"><span id="MathJax-Span-337" class="mrow"><span id="MathJax-Span-338" class="texatom"><span id="MathJax-Span-339" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-340" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-341" class="mo">=</span><span id="MathJax-Span-342" class="msubsup"><span style="display: inline-block; position: relative; width: 0.677em; height: 0px;"><span style="position: absolute; clip: rect(3.544em, 1000.34em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-343" class="mi">μ</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.82em; left: 0.372em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-344" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.268em, 1001.4em, 4.343em, -1000em); top: -5.299em; left: 0.04em;"><span id="MathJax-Span-345" class="texatom"><span id="MathJax-Span-346" class="mrow"><span id="MathJax-Span-347" class="texatom"><span id="MathJax-Span-348" class="mrow"><span id="MathJax-Span-349" class="msubsup"><span style="display: inline-block; position: relative; width: 0.637em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.33em, 4.147em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-350" class="mi">ν</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.332em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-351" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-352" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-353" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-354" class="mo">{</span><span id="MathJax-Span-355" class="msubsup"><span style="display: inline-block; position: relative; width: 1.751em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.41em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-356" class="mi">e</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.385em, 1000.27em, 4.281em, -1000em); top: -4.481em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-357" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.3em, 1001.31em, 4.28em, -1000em); top: -3.609em; left: 0.444em;"><span id="MathJax-Span-358" class="texatom"><span id="MathJax-Span-359" class="mrow"><span id="MathJax-Span-360" class="texatom"><span id="MathJax-Span-361" class="mrow"><span id="MathJax-Span-362" class="msubsup"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.534em, 1000.33em, 4.28em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-363" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.195em; left: 0.356em;"><span id="MathJax-Span-364" class="texatom"><span id="MathJax-Span-365" class="mrow"><span id="MathJax-Span-366" class="texatom"><span id="MathJax-Span-367" class="mrow"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-368" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.009em;"></span></span><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-369" class="mo">+</span><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-370" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-371" class="msubsup"><span style="display: inline-block; position: relative; width: 2.211em; height: 0px;"><span style="position: absolute; clip: rect(3.166em, 1000.38em, 4.316em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-372" class="mo">}</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.48em;"><span id="MathJax-Span-373" class="texatom"><span id="MathJax-Span-374" class="mrow"><span id="MathJax-Span-375" class="texatom"><span id="MathJax-Span-376" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-377" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-378" class="mo">∈</span><span id="MathJax-Span-379" class="msubsup"><span style="display: inline-block; position: relative; width: 0.974em; height: 0px;"><span style="position: absolute; clip: rect(3.384em, 1000.35em, 4.147em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-380" class="mi">J<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.033em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.314em;"><span id="MathJax-Span-381" class="texatom"><span id="MathJax-Span-382" class="mrow"><span id="MathJax-Span-383" class="texatom"><span id="MathJax-Span-384" class="mrow"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-385" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.009em;"></span></span><span style="font-size: 50%; font-family: STIXGeneral;" id="MathJax-Span-386" class="mo">,</span><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-387" class="mi">p</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.042em; border-left: 0px solid; width: 0px; height: 4.558em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML MJX_Assistive_MathML_Block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>,</mo><mi>m</mi></mrow></mrow></msub><mo>=</mo><munder><mo>⨂</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>p</mi><mo>∈</mo><mi>P</mi></mrow></mrow></munder><munderover><mo>⨂</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><msub><mi>μ</mi><mi>p</mi></msub></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msub><mi>ν</mi><mi>p</mi></msub><mo>−</mo><mn>1</mn></mrow></mrow></munderover><mo fence="false" stretchy="false">{</mo><msubsup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mrow></msup></mrow></mrow><mi>j</mi></msubsup><msub><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>∈</mo><msub><mi>J</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>,</mo><mi>p</mi></mrow></mrow></msub></mrow></mrow></msub></math></span></span></div><script id="MathJax-Element-41" type="math/tex; mode=display">
-B_{{n,m}} = \bigotimes_{{p \in P}}
- \bigotimes_{{k = \mu_p}}^{{\nu_p-1}}
- \{ e_{{p^{{k+1}}}}^j \}_{{j \in J_{{k,p}}}}
-</script></p>
-
-<p>where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>J</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>,</mo><mi>p</mi></mrow></mrow></msub><mo>=</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-42-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.078em; display: inline-block;" role="math" id="MathJax-Span-388" class="math"><span style="display: inline-block; position: relative; width: 2.356em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1002.31em, 3.083em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-389" class="mrow"><span id="MathJax-Span-390" class="msubsup"><span style="display: inline-block; position: relative; width: 1.378em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.49em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-391" class="mi">J<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-392" class="texatom"><span id="MathJax-Span-393" class="mrow"><span id="MathJax-Span-394" class="texatom"><span id="MathJax-Span-395" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-396" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-397" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-398" class="mi">p</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-399" class="mo">=</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.446em; border-left: 0px solid; width: 0px; height: 1.357em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>J</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>,</mo><mi>p</mi></mrow></mrow></msub><mo>=</mo></math></span></span><script id="MathJax-Element-42" type="math/tex">J_{{k,p}} =</script></span></p>
-
-<div class="pcenter"><table class="GAPDocTablenoborder">
-<tbody><tr>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo fence="false" stretchy="false">}</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-43-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.891em; display: inline-block;" role="math" id="MathJax-Span-400" class="math"><span style="display: inline-block; position: relative; width: 1.442em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.82em, 1001.34em, 2.969em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-401" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-402" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-403" class="mn">0</span><span style="font-family: STIXGeneral;" id="MathJax-Span-404" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.298em; border-left: 0px solid; width: 0px; height: 1.244em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script id="MathJax-Element-43" type="math/tex">\{ 0 \}</script></span></td>
-<td class="tdcenter">;</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>p</mi><mo>=</mo><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-44-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 6.391em; display: inline-block;" role="math" id="MathJax-Span-405" class="math"><span style="display: inline-block; position: relative; width: 4.904em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1004.88em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-406" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-407" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-408" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-409" class="mn">0</span><span style="font-family: STIXGeneral;" id="MathJax-Span-410" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-411" class="mi">p</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-412" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-413" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.279em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>p</mi><mo>=</mo><mn>2</mn></math></span></span><script id="MathJax-Element-44" type="math/tex">k = 0, p = 2</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-45-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.078em; display: inline-block;" role="math" id="MathJax-Span-414" class="math"><span style="display: inline-block; position: relative; width: 2.356em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.82em, 1002.26em, 2.969em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-415" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-416" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-417" class="mn">0</span><span style="font-family: STIXGeneral;" id="MathJax-Span-418" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-419" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-420" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.298em; border-left: 0px solid; width: 0px; height: 1.244em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script id="MathJax-Element-45" type="math/tex">\{ 0, 1 \}</script></span></td>
-<td class="tdcenter">;</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mi>p</mi><mo>=</mo><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-46-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 6.391em; display: inline-block;" role="math" id="MathJax-Span-421" class="math"><span style="display: inline-block; position: relative; width: 4.904em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1004.88em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-422" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-423" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-424" class="mo">></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-425" class="mn">0</span><span style="font-family: STIXGeneral;" id="MathJax-Span-426" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-427" class="mi">p</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-428" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-429" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.279em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>p</mi><mo>=</mo><mn>2</mn></math></span></span><script id="MathJax-Element-46" type="math/tex">k > 0, p = 2</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><mi>p</mi><mo>&#x2212;</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-47-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.328em; display: inline-block;" role="math" id="MathJax-Span-430" class="math"><span style="display: inline-block; position: relative; width: 5.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.82em, 1005.53em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-431" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-432" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-433" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-434" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-435" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-436" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-437" class="mi">p</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-438" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-439" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-440" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.276em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script id="MathJax-Element-47" type="math/tex">\{ 1, \ldots, p-1 \}</script></span></td>
-<td class="tdcenter">;</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>p</mi><mo>&#x2260;</mo><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-48-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 6.391em; display: inline-block;" role="math" id="MathJax-Span-441" class="math"><span style="display: inline-block; position: relative; width: 4.904em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1004.88em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-442" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-443" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-444" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-445" class="mn">0</span><span style="font-family: STIXGeneral;" id="MathJax-Span-446" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-447" class="mi">p</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-448" class="mo">≠</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-449" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.279em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>p</mi><mo>≠</mo><mn>2</mn></math></span></span><script id="MathJax-Element-48" type="math/tex">k = 0, p \neq 2</script></span></td>
-</tr>
-<tr>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mo>&#x2212;</mo><mo stretchy="false">(</mo><mi>p</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><mo stretchy="false">(</mo><mi>p</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo fence="false" stretchy="false">}</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-49-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 14.141em; display: inline-block;" role="math" id="MathJax-Span-450" class="math"><span style="display: inline-block; position: relative; width: 10.865em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.82em, 1010.77em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-451" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-452" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-453" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-454" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-455" class="mi">p</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-456" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-457" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-458" class="mo">)</span><span id="MathJax-Span-459" class="texatom"><span id="MathJax-Span-460" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-461" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-462" class="mn">2</span><span style="font-family: STIXGeneral;" id="MathJax-Span-463" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-464" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-465" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-466" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-467" class="mi">p</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-468" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-469" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-470" class="mo">)</span><span id="MathJax-Span-471" class="texatom"><span id="MathJax-Span-472" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-473" class="mo">/</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-474" class="mn">2</span><span style="font-family: STIXGeneral;" id="MathJax-Span-475" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.276em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mo>−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script id="MathJax-Element-49" type="math/tex">\{ -(p-1)/2, \ldots, (p-1)/2 \}</script></span></td>
-<td class="tdcenter">;</td>
-<td class="tdleft"><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mi>p</mi><mo>&#x2260;</mo><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-50-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 6.391em; display: inline-block;" role="math" id="MathJax-Span-476" class="math"><span style="display: inline-block; position: relative; width: 4.904em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1004.88em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-477" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-478" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-479" class="mo">></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-480" class="mn">0</span><span style="font-family: STIXGeneral;" id="MathJax-Span-481" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-482" class="mi">p</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-483" class="mo">≠</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-484" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.279em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>p</mi><mo>≠</mo><mn>2</mn></math></span></span><script id="MathJax-Element-50" type="math/tex">k > 0, p \neq 2</script></span></td>
-</tr>
-</tbody></table><br><p> </p><br>
-</div>
-
-<p><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>,</mo><mn>1</mn></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-51-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.078em; display: inline-block;" role="math" id="MathJax-Span-485" class="math"><span style="display: inline-block; position: relative; width: 1.587em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1001.59em, 2.846em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-486" class="mrow"><span id="MathJax-Span-487" class="msubsup"><span style="display: inline-block; position: relative; width: 1.57em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.59em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-488" class="mi">B</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span id="MathJax-Span-489" class="texatom"><span id="MathJax-Span-490" class="mrow"><span id="MathJax-Span-491" class="texatom"><span id="MathJax-Span-492" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-493" class="mi">n</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-494" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-495" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.387em; border-left: 0px solid; width: 0px; height: 1.299em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>,</mo><mn>1</mn></mrow></mrow></msub></math></span></span><script id="MathJax-Element-51" type="math/tex">B_{{n,1}}</script></span> is equal to the basis of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-52-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-496" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-497" class="mrow"><span id="MathJax-Span-498" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-499" class="texatom"><span id="MathJax-Span-500" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-501" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-502" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-52" type="math/tex">â„š_n</script></span> over the rationals which is introduced in <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBZum89">[Zum89]</a>.
- Also the conversion of arbitrary sums of roots of unity into its basis
-representation, and the reduction to the minimal cyclotomic field are
-described in this thesis. (Note that the notation here is slightly
-different from that there.)</p>
-
-<p><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>,</mo><mi>m</mi></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-53-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.266em; display: inline-block;" role="math" id="MathJax-Span-503" class="math"><span style="display: inline-block; position: relative; width: 1.731em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1001.73em, 2.846em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-504" class="mrow"><span id="MathJax-Span-505" class="msubsup"><span style="display: inline-block; position: relative; width: 1.727em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.59em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-506" class="mi">B</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span id="MathJax-Span-507" class="texatom"><span id="MathJax-Span-508" class="mrow"><span id="MathJax-Span-509" class="texatom"><span id="MathJax-Span-510" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-511" class="mi">n</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-512" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-513" class="mi">m</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.387em; border-left: 0px solid; width: 0px; height: 1.299em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>,</mo><mi>m</mi></mrow></mrow></msub></math></span></span><script id="MathJax-Element-53" type="math/tex">B_{{n,m}}</script></span> consists of roots of unity, it is an integral basis (that is, exactly the integral elements in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-54-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-514" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-515" class="mrow"><span id="MathJax-Span-516" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-517" class="texatom"><span id="MathJax-Span-518" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-519" class="mo">â„š</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-520" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>â„š</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-54" type="math/tex">â„š_n</script></span> have integral coefficients w.r.t. <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>,</mo><mi>m</mi></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-55-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.266em; display: inline-block;" role="math" id="MathJax-Span-521" class="math"><span style="display: inline-block; position: relative; width: 1.731em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1001.73em, 2.846em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-522" class="mrow"><span id="MathJax-Span-523" class="msubsup"><span style="display: inline-block; position: relative; width: 1.727em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.59em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-524" class="mi">B</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span id="MathJax-Span-525" class="texatom"><span id="MathJax-Span-526" class="mrow"><span id="MathJax-Span-527" class="texatom"><span id="MathJax-Span-528" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-529" class="mi">n</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-530" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-531" class="mi">m</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.387em; border-left: 0px solid; width: 0px; height: 1.299em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>,</mo><mi>m</mi></mrow></mrow></msub></math></span></span><script id="MathJax-Element-55" type="math/tex">B_{{n,m}}</script></span>, cf. <code class="func">IsIntegralCyclotomic</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X869750DA81EA0E67"><span class="RefLink">18.1-4</span></a>)), it is a normal basis for squarefree <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-56-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-532" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-533" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-534" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-56" type="math/tex">n</script></span> and closed under complex conjugation for odd <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-57-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-535" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-536" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-537" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-57" type="math/tex">n</script></span>.</p>
-
-<p><em>Note:</em> For <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">n</mtext></mrow><mo>&#x2261;</mo><mn>2</mn><mspace width="0.444em" /><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em" /><mn>4</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-58-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.078em; display: inline-block;" role="math" id="MathJax-Span-538" class="math"><span style="display: inline-block; position: relative; width: 6.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1006.15em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-539" class="mrow"><span id="MathJax-Span-540" class="texatom"><span id="MathJax-Span-541" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-542" class="mtext">n</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-543" class="mo">≡</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-544" class="mn">2</span><span id="MathJax-Span-545" class="TeXmathchoice"><span style="height: 0em; vertical-align: 0em; width: 0.5em; display: inline-block; overflow: hidden;" id="MathJax-Span-546" class="mspace"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-547" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-548" class="mi">mod</span><span style="height: 0em; vertical-align: 0em; width: 0.375em; display: inline-block; overflow: hidden;" id="MathJax-Span-549" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-550" class="mn">4</span><span style="font-family: STIXGeneral;" id="MathJax-Span-551" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">n</mtext></mrow><mo>≡</mo><mn>2</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>4</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-58" type="math/tex">\textit{n} \equiv 2 \pmod 4</script></span>, we have <code class="code">ZumbroichBase(<var class="Arg">n</var>, 1) = 2 * ZumbroichBase(<var class="Arg">n</var>/2, 1)</code> and <code class="code">List( ZumbroichBase(<var class="Arg">n</var>, 1), x -> E(<var class="Arg">n</var>)^x ) = List( ZumbroichBase(<var class="Arg">n</var>/2, 1), x -> E(<var class="Arg">n</var>/2)^x )</code>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">ZumbroichBase( 15, 1 ); ZumbroichBase( 12, 3 );</span>
-[ 1, 2, 4, 7, 8, 11, 13, 14 ]
-[ 0, 3 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">ZumbroichBase( 10, 2 ); ZumbroichBase( 32, 4 );</span>
-[ 2, 4, 6, 8 ]
-[ 0, 1, 2, 3, 4, 5, 6, 7 ]
-</pre></div>
-
-<p><a id="X87DB9C2C858B722A" name="X87DB9C2C858B722A"></a></p>
-
-<h5>60.3-2 LenstraBase</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ LenstraBase</code>( <var class="Arg">n</var>, <var class="Arg">stabilizer</var>, <var class="Arg">super</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">n</var> and <var class="Arg">m</var> be positive integers such that <var class="Arg">m</var> divides <var class="Arg">n</var>, <var class="Arg">stabilizer</var> be a list of prime residues modulo <var class="Arg">n</var>, which describes a subfield of the <var class="Arg">n</var>-th cyclotomic field (see <code class="func">GaloisStabilizer</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X87E7313D8070B9CC"><span class="RefLink">60.2-5</span></a>)), and <var class="Arg">super</var> be a list representing a supergroup of the group given by <var class="Arg">stabilizer</var>.</p>
-
-<p><code class="func">LenstraBase</code> returns a list <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>b</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>2</mn></msub><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><msub><mi>b</mi><mi>k</mi></msub><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-59-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.703em; display: inline-block;" role="math" id="MathJax-Span-552" class="math"><span style="display: inline-block; position: relative; width: 5.913em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1005.83em, 2.946em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-553" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-554" class="mo">[</span><span id="MathJax-Span-555" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-556" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-557" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-558" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-559" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-560" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-561" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-562" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-563" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-564" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-565" class="msubsup"><span style="display: inline-block; position: relative; width: 0.901em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-566" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-567" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-568" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>b</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>b</mi><mi>k</mi></msub><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-59" type="math/tex">[ b_1, b_2, \ldots, b_k ]</script></span> of lists, each <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-60-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-569" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.77em, 2.754em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-570" class="mrow"><span id="MathJax-Span-571" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-572" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-573" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub></math></span></span><script id="MathJax-Element-60" type="math/tex">b_i</script></span> consisting of integers such that the elements <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>&#x2208;</mo><msub><mi>b</mi><mi>i</mi></msub></mrow></mrow></munder></math>" style="position: relative;" tabindex="0" id="MathJax-Element-61-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.891em; display: inline-block;" role="math" id="MathJax-Span-574" class="math"><span style="display: inline-block; position: relative; width: 2.212em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.547em, 1002.21em, 3.039em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-575" class="mrow"><span id="MathJax-Span-576" class="munderover"><span style="display: inline-block; position: relative; width: 2.216em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-577" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.694em; left: 0.914em;"><span id="MathJax-Span-578" class="texatom"><span id="MathJax-Span-579" class="mrow"><span id="MathJax-Span-580" class="texatom"><span id="MathJax-Span-581" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-582" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-583" class="mo">∈</span><span id="MathJax-Span-584" class="msubsup"><span style="display: inline-block; position: relative; width: 0.546em; height: 0px;"><span style="position: absolute; clip: rect(3.363em, 1000.33em, 4.142em, -1000em); top: -3.99em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-585" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.884em; left: 0.354em;"><span style="font-size: 50%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-586" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.638em; border-left: 0px solid; width: 0px; height: 1.69em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>∈</mo><msub><mi>b</mi><mi>i</mi></msub></mrow></mrow></munder></math></span></span><script id="MathJax-Element-61" type="math/tex">\sum_{{j \in b_i}} </script></span><code class="code">E(n)</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi></mi><mi>j</mi></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-62-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-587" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.484em, 1000.29em, 2.596em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-588" class="mrow"><span id="MathJax-Span-589" class="msubsup"><span style="display: inline-block; position: relative; width: 0.272em; height: 0px;"><span style="position: absolute; clip: rect(3.846em, 1000em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-590" class="mi"></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-591" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.196em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi></mi><mi>j</mi></msup></math></span></span><script id="MathJax-Element-62" type="math/tex">^j</script></span> form a basis of the abelian number field <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stabilizer</var> )</code>, as a vector space over the <var class="Arg">m</var>-th cyclotomic field (see <code class="func">AbelianNumberField</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>)).</p>
-
-<p>This basis is an integral basis, that is, exactly the integral elements in <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stabilizer</var> )</code> have integral coefficients. (For details about this basis, see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBBre97">[Bre97]</a>.)</p>
-
-<p>If possible then the result is chosen such that the group described by <var class="Arg">super</var> acts on it, consistently with the action of <var class="Arg">stabilizer</var>, i.e., each orbit of <var class="Arg">super</var> is a union of orbits of <var class="Arg">stabilizer</var>. (A usual case is <var class="Arg">super</var><code class="code"> = </code><var class="Arg">stabilizer</var>, so there is no additional condition.</p>
-
-<p><em>Note:</em> The <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-63-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-592" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.77em, 2.754em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-593" class="mrow"><span id="MathJax-Span-594" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-595" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-596" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub></math></span></span><script id="MathJax-Element-63" type="math/tex">b_i</script></span> are in general not sets, since for <code class="code"><var class="Arg">stabilizer</var> = <var class="Arg">super</var></code>, the first entry is always an element of <code class="code">ZumbroichBase( <var class="Arg">n</var>, <var class="Arg">m</var> )</code>; this property is used by <code class="func">NF</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>) and <code class="func">Coefficients</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap61_mj.html#X80B32F667BF6AFD8"><span class="RefLink">61.6-3</span></a>) (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p>
-
-<p><var class="Arg">stabilizer</var> must not contain the stabilizer of a proper cyclotomic subfield of the <var class="Arg">n</var>-th cyclotomic field, i.e., the result must describe a basis for a field with conductor <var class="Arg">n</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 24, [ 1, 19 ], [ 1, 19 ], 1 );</span>
-[ [ 1, 19 ], [ 8 ], [ 11, 17 ], [ 16 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 24, [ 1, 19 ], [ 1, 5, 19, 23 ], 1 );</span>
-[ [ 1, 19 ], [ 5, 23 ], [ 8 ], [ 16 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 15, [ 1, 4 ], PrimeResidues( 15 ), 1 );</span>
-[ [ 1, 4 ], [ 2, 8 ], [ 7, 13 ], [ 11, 14 ] ]
-</pre></div>
-
-<p>The first two results describe two bases of the field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mn>6</mn></mrow></msqrt><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-64-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.078em; display: inline-block;" role="math" id="MathJax-Span-597" class="math"><span style="display: inline-block; position: relative; width: 3.125em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.673em, 1003.08em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-598" class="mrow"><span id="MathJax-Span-599" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-600" class="texatom"><span id="MathJax-Span-601" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-602" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-603" class="mn">3</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-604" class="mo">(</span><span id="MathJax-Span-605" class="msqrt"><span style="display: inline-block; position: relative; width: 1.237em; height: 0px;"><span style="position: absolute; clip: rect(3.162em, 1000.47em, 4.149em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-606" class="mrow"><span id="MathJax-Span-607" class="texatom"><span id="MathJax-Span-608" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-609" class="mn">6</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1000.5em, 3.365em, -1000em); top: -3.997em; left: 0.737em;"><span style="font-family: STIXGeneral;">‾</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.874em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-610" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.43em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mn>6</mn></mrow></msqrt><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-64" type="math/tex">ℚ_3(\sqrt{{6}})</script></span>, the third result describes a normal basis of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mn>5</mn></mrow></msqrt><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-65-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.078em; display: inline-block;" role="math" id="MathJax-Span-611" class="math"><span style="display: inline-block; position: relative; width: 3.125em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.671em, 1003.08em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-612" class="mrow"><span id="MathJax-Span-613" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-614" class="texatom"><span id="MathJax-Span-615" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-616" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-617" class="mn">3</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-618" class="mo">(</span><span id="MathJax-Span-619" class="msqrt"><span style="display: inline-block; position: relative; width: 1.237em; height: 0px;"><span style="position: absolute; clip: rect(3.158em, 1000.44em, 4.149em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-620" class="mrow"><span id="MathJax-Span-621" class="texatom"><span id="MathJax-Span-622" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-623" class="mn">5</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1000.5em, 3.365em, -1000em); top: -3.999em; left: 0.737em;"><span style="font-family: STIXGeneral;">‾</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.876em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-624" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.433em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mn>5</mn></mrow></msqrt><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-65" type="math/tex">ℚ_3(\sqrt{{5}})</script></span>.</p>
-
-<p><a id="X7E4AB4B17C7BA10C" name="X7E4AB4B17C7BA10C"></a></p>
-
-<h4>60.4 <span class="Heading">Galois Groups of Abelian Number Fields</span></h4>
-
-<p>The field automorphisms of the cyclotomic field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-66-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-625" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-626" class="mrow"><span id="MathJax-Span-627" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-628" class="texatom"><span id="MathJax-Span-629" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-630" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-631" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-66" type="math/tex">ℚ_n</script></span> (see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>) are given by the linear maps <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2217;</mo><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-67-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-632" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.96em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-633" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-634" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-635" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∗</mo><mi>k</mi></math></span></span><script id="MathJax-Element-67" type="math/tex">*k</script></span> on <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-68-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-636" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-637" class="mrow"><span id="MathJax-Span-638" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-639" class="texatom"><span id="MathJax-Span-640" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-641" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-642" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-68" type="math/tex">ℚ_n</script></span> that are defined by <code class="code">E</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mi>k</mi></mrow></mrow></msup><mo>=</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-69-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.766em; display: inline-block;" role="math" id="MathJax-Span-643" class="math"><span style="display: inline-block; position: relative; width: 2.885em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.654em, 1002.84em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-644" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-645" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-646" class="mi">n</span><span id="MathJax-Span-647" class="msubsup"><span style="display: inline-block; position: relative; width: 1.104em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-648" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span id="MathJax-Span-649" class="texatom"><span id="MathJax-Span-650" class="mrow"><span id="MathJax-Span-651" class="texatom"><span id="MathJax-Span-652" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-653" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-654" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-655" class="mo">=</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.455em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mi>k</mi></mrow></mrow></msup><mo>=</mo></math></span></span><script id="MathJax-Element-69" type="math/tex">(n)^{{*k}} = </script></span><code class="code">E</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo><mi>k</mi></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-70-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.016em; display: inline-block;" role="math" id="MathJax-Span-656" class="math"><span style="display: inline-block; position: relative; width: 1.538em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.654em, 1001.54em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-657" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-658" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-659" class="mi">n</span><span id="MathJax-Span-660" class="msubsup"><span style="display: inline-block; position: relative; width: 0.734em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-661" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-662" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.455em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo><mi>k</mi></msup></math></span></span><script id="MathJax-Element-70" type="math/tex">(n)^k</script></span>, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>&#x2264;</mo><mi>k</mi><mo>&lt;</mo><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-71-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.203em; display: inline-block;" role="math" id="MathJax-Span-663" class="math"><span style="display: inline-block; position: relative; width: 3.99em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1003.96em, 2.891em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-664" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-665" class="mn">1</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-666" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-667" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-668" class="mo"><</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-669" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.196em; border-left: 0px solid; width: 0px; height: 1.147em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>n</mi></math></span></span><script id="MathJax-Element-71" type="math/tex">1 \leq k < n</script></span> and <code class="code">Gcd</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-72-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.953em; display: inline-block;" role="math" id="MathJax-Span-670" class="math"><span style="display: inline-block; position: relative; width: 3.798em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1003.69em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-671" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-672" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-673" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-674" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-675" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-676" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-677" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-678" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></span></span><script id="MathJax-Element-72" type="math/tex">( n, k ) = 1</script></span> hold (see <code class="func">GaloisCyc</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X79EE9097783128C4"><span class="RefLink">18.5-1</span></a>)). Note that this action is <em>not</em> equal to exponentiation of cyclotomics, i.e., for general cyclotomics <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-73-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.516em; display: inline-block;" role="math" id="MathJax-Span-679" class="math"><span style="display: inline-block; position: relative; width: 0.385em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.38em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-680" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-681" class="mi">z</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 0.787em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></span></span><script id="MathJax-Element-73" type="math/tex">z</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mi>k</mi></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-74-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.516em; display: inline-block;" role="math" id="MathJax-Span-682" class="math"><span style="display: inline-block; position: relative; width: 1.154em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.462em, 1001.15em, 2.677em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-683" class="mrow"><span id="MathJax-Span-684" class="msubsup"><span style="display: inline-block; position: relative; width: 1.16em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-685" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-686" class="texatom"><span id="MathJax-Span-687" class="mrow"><span id="MathJax-Span-688" class="texatom"><span id="MathJax-Span-689" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-690" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-691" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.33em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mi>k</mi></mrow></mrow></msup></math></span></span><script id="MathJax-Element-74" type="math/tex">z^{{*k}}</script></span> is different from <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>k</mi></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-75-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-692" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.462em, 1000.77em, 2.677em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-693" class="mrow"><span id="MathJax-Span-694" class="msubsup"><span style="display: inline-block; position: relative; width: 0.79em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-695" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-696" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 1.33em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>k</mi></msup></math></span></span><script id="MathJax-Element-75" type="math/tex">z^k</script></span>.</p>
-
-<p>(In <strong class="pkg">GAP</strong>, the image of a cyclotomic <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-76-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.516em; display: inline-block;" role="math" id="MathJax-Span-697" class="math"><span style="display: inline-block; position: relative; width: 0.385em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.38em, 2.869em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-698" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-699" class="mi">z</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.168em; border-left: 0px solid; width: 0px; height: 0.787em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></span></span><script id="MathJax-Element-76" type="math/tex">z</script></span> under <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2217;</mo><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-77-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-700" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.96em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-701" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-702" class="mo">∗</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-703" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∗</mo><mi>k</mi></math></span></span><script id="MathJax-Element-77" type="math/tex">*k</script></span> can be computed as <code class="code">GaloisCyc( </code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>,</mo><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-78-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.641em; display: inline-block;" role="math" id="MathJax-Span-704" class="math"><span style="display: inline-block; position: relative; width: 1.25em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1001.25em, 2.929em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-705" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-706" class="mi">z</span><span style="font-family: STIXGeneral;" id="MathJax-Span-707" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-708" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.246em; border-left: 0px solid; width: 0px; height: 1.196em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>,</mo><mi>k</mi></math></span></span><script id="MathJax-Element-78" type="math/tex">z, k</script></span><code class="code"> )</code>.)</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">( E(5) + E(5)^4 )^2; GaloisCyc( E(5) + E(5)^4, 2 );</span>
--2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
-E(5)^2+E(5)^3
-</pre></div>
-
-<p>For <code class="code">Gcd</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>&#x2260;</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-79-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.953em; display: inline-block;" role="math" id="MathJax-Span-709" class="math"><span style="display: inline-block; position: relative; width: 3.798em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1003.69em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-710" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-711" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-712" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-713" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-714" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-715" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-716" class="mo">≠</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-717" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>1</mn></math></span></span><script id="MathJax-Element-79" type="math/tex">( n, k ) \neq 1</script></span>, the map <code class="code">E</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">&#x21A6;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-80-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.141em; display: inline-block;" role="math" id="MathJax-Span-718" class="math"><span style="display: inline-block; position: relative; width: 2.404em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.34em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-719" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-720" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-721" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-722" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-723" class="mo">↦</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">↦</mo></math></span></span><script id="MathJax-Element-80" type="math/tex">(n) \mapsto</script></span> <code class="code">E</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo><mi>k</mi></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-81-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.016em; display: inline-block;" role="math" id="MathJax-Span-724" class="math"><span style="display: inline-block; position: relative; width: 1.538em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.654em, 1001.54em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-725" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-726" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-727" class="mi">n</span><span id="MathJax-Span-728" class="msubsup"><span style="display: inline-block; position: relative; width: 0.734em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-729" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-730" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.455em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><msup><mo stretchy="false">)</mo><mi>k</mi></msup></math></span></span><script id="MathJax-Element-81" type="math/tex">(n)^k</script></span> does <em>not</em> define a field automorphism of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-82-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.578em; display: inline-block;" role="math" id="MathJax-Span-731" class="math"><span style="display: inline-block; position: relative; width: 1.202em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.632em, 1001.2em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-732" class="mrow"><span id="MathJax-Span-733" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-734" class="texatom"><span id="MathJax-Span-735" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-736" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-737" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-82" type="math/tex">ℚ_n</script></span> but only a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-83-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-738" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.72em, 2.853em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-739" class="mrow"><span id="MathJax-Span-740" class="texatom"><span id="MathJax-Span-741" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-742" class="mo">ℚ</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.147em; border-left: 0px solid; width: 0px; height: 1.088em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow></math></span></span><script id="MathJax-Element-83" type="math/tex">ℚ</script></span>-linear map.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 );</span>
-2
--6
-</pre></div>
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-<p><a id="X7B55A90582E818F3" name="X7B55A90582E818F3"></a></p>
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-<h5>60.4-1 GaloisGroup</h5>
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-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ GaloisGroup</code>( <var class="Arg">F</var> )</td><td class="tdright">( method )</td></tr></tbody></table></div>
-<p>The Galois group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>a</mi><mi>l</mi><mo stretchy="false">(</mo><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-84-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.953em; display: inline-block;" role="math" id="MathJax-Span-743" class="math"><span style="display: inline-block; position: relative; width: 4.567em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1004.52em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-744" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-745" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-746" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-747" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-748" class="mo">(</span><span id="MathJax-Span-749" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-750" class="texatom"><span id="MathJax-Span-751" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-752" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-753" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-754" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-755" class="texatom"><span id="MathJax-Span-756" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-757" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-758" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>a</mi><mi>l</mi><mo stretchy="false">(</mo><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mi>n</mi></msub><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-84" type="math/tex">Gal( ℚ_n, ℚ )</script></span> of the field extension <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-85-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.953em; display: inline-block;" role="math" id="MathJax-Span-759" class="math"><span style="display: inline-block; position: relative; width: 2.26em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.21em, 2.945em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-760" class="mrow"><span id="MathJax-Span-761" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-762" class="texatom"><span id="MathJax-Span-763" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-764" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-765" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-766" class="texatom"><span id="MathJax-Span-767" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-768" class="mo">/</span></span></span><span id="MathJax-Span-769" class="texatom"><span id="MathJax-Span-770" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-771" class="mo">ℚ</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mi>n</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow></math></span></span><script id="MathJax-Element-85" type="math/tex">ℚ_n / ℚ</script></span> is isomorphic to the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-86-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.328em; display: inline-block;" role="math" id="MathJax-Span-772" class="math"><span style="display: inline-block; position: relative; width: 3.317em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.804em, 1003.32em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-773" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-774" class="mo">(</span><span id="MathJax-Span-775" class="texatom"><span id="MathJax-Span-776" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-777" class="mo">ℤ</span></span></span><span id="MathJax-Span-778" class="texatom"><span id="MathJax-Span-779" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-780" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-781" class="mi">n</span><span id="MathJax-Span-782" class="texatom"><span id="MathJax-Span-783" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-784" class="mo">ℤ</span></span></span><span id="MathJax-Span-785" class="msubsup"><span style="display: inline-block; position: relative; width: 0.778em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-786" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span id="MathJax-Span-787" class="texatom"><span id="MathJax-Span-788" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-789" class="mo">∗</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.26em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo></mrow></msup></math></span></span><script id="MathJax-Element-86" type="math/tex">(ℤ / n ℤ)^{*}</script></span> of prime residues modulo <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-87-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-790" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-791" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-792" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-87" type="math/tex">n</script></span>, via the isomorphism <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo></mrow></msup><mo stretchy="false">&#x2192;</mo><mi>G</mi><mi>a</mi><mi>l</mi><mo stretchy="false">(</mo><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-88-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 12.266em; display: inline-block;" role="math" id="MathJax-Span-793" class="math"><span style="display: inline-block; position: relative; width: 9.423em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.804em, 1009.38em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-794" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-795" class="mo">(</span><span id="MathJax-Span-796" class="texatom"><span id="MathJax-Span-797" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-798" class="mo">ℤ</span></span></span><span id="MathJax-Span-799" class="texatom"><span id="MathJax-Span-800" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-801" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-802" class="mi">n</span><span id="MathJax-Span-803" class="texatom"><span id="MathJax-Span-804" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-805" class="mo">ℤ</span></span></span><span id="MathJax-Span-806" class="msubsup"><span style="display: inline-block; position: relative; width: 0.778em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-807" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span id="MathJax-Span-808" class="texatom"><span id="MathJax-Span-809" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-810" class="mo">∗</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-811" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-812" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-813" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-814" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-815" class="mo">(</span><span id="MathJax-Span-816" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-817" class="texatom"><span id="MathJax-Span-818" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-819" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-820" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-821" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-822" class="texatom"><span id="MathJax-Span-823" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-824" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-825" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.26em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo></mrow></msup><mo stretchy="false">→</mo><mi>G</mi><mi>a</mi><mi>l</mi><mo stretchy="false">(</mo><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mi>n</mi></msub><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-88" type="math/tex">(ℤ / n ℤ)^{*} \rightarrow Gal( ℚ_n, ℚ )</script></span> that is defined by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>+</mo><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow><mo stretchy="false">&#x21A6;</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">&#x21A6;</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-89-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 10.578em; display: inline-block;" role="math" id="MathJax-Span-826" class="math"><span style="display: inline-block; position: relative; width: 8.125em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.654em, 1008.08em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-827" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-828" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-829" class="mo">+</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-830" class="mi">n</span><span id="MathJax-Span-831" class="texatom"><span id="MathJax-Span-832" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-833" class="mo">ℤ</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-834" class="mo">↦</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-835" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-836" class="mi">z</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-837" class="mo">↦</span><span style="padding-left: 0.313em;" id="MathJax-Span-838" class="msubsup"><span style="display: inline-block; position: relative; width: 1.16em; height: 0px;"><span style="position: absolute; clip: rect(3.418em, 1000.38em, 4.216em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-839" class="mi">z</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.389em;"><span id="MathJax-Span-840" class="texatom"><span id="MathJax-Span-841" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-842" class="mo">∗</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-843" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-844" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.455em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>+</mo><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow><mo stretchy="false">↦</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">↦</mo><msup><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-89" type="math/tex">k + n ℤ \mapsto ( z \mapsto z^{*k} )</script></span>.</p>
-
-<p>The Galois group of the field extension <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>L</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-90-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.703em; display: inline-block;" role="math" id="MathJax-Span-845" class="math"><span style="display: inline-block; position: relative; width: 2.067em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.07em, 2.945em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-846" class="mrow"><span id="MathJax-Span-847" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-848" class="texatom"><span id="MathJax-Span-849" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-850" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-851" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-852" class="texatom"><span id="MathJax-Span-853" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-854" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-855" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mi>n</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>L</mi></math></span></span><script id="MathJax-Element-90" type="math/tex">ℚ_n / L</script></span> with any abelian number field <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>&#x2286;</mo><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-91-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.016em; display: inline-block;" role="math" id="MathJax-Span-856" class="math"><span style="display: inline-block; position: relative; width: 3.077em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.08em, 2.945em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-857" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-858" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-859" class="mo">⊆</span><span style="padding-left: 0.313em;" id="MathJax-Span-860" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-861" class="texatom"><span id="MathJax-Span-862" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-863" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-864" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>⊆</mo><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mi>n</mi></msub></math></span></span><script id="MathJax-Element-91" type="math/tex">L \subseteq ℚ_n</script></span> is simply the factor group of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>a</mi><mi>l</mi><mo stretchy="false">(</mo><msub><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mi>n</mi></msub><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x211A;</mo></mrow><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-92-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.953em; display: inline-block;" role="math" id="MathJax-Span-865" class="math"><span style="display: inline-block; position: relative; width: 4.567em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1004.52em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-866" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-867" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-868" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-869" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-870" class="mo">(</span><span id="MathJax-Span-871" class="msubsup"><span style="display: inline-block; position: relative; width: 1.194em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.72em, 4.2em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-872" class="texatom"><span id="MathJax-Span-873" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-874" class="mo">ℚ</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.765em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-875" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-876" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-877" class="texatom"><span id="MathJax-Span-878" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-879" class="mo">ℚ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-880" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.243em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>a</mi><mi>l</mi><mo stretchy="false">(</mo><msub><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mi>n</mi></msub><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mo>ℚ</mo></mrow><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-92" type="math/tex">Gal( ℚ_n, ℚ )</script></span> modulo the stabilizer of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-93-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.766em; display: inline-block;" role="math" id="MathJax-Span-881" class="math"><span style="display: inline-block; position: relative; width: 0.577em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.58em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-882" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-883" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math></span></span><script id="MathJax-Element-93" type="math/tex">L</script></span>, and the Galois group of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>L</mi><mo>&#x2032;</mo></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-94-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.328em; display: inline-block;" role="math" id="MathJax-Span-884" class="math"><span style="display: inline-block; position: relative; width: 1.779em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.738em, 1001.78em, 2.802em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-885" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-886" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-887" class="texatom"><span id="MathJax-Span-888" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-889" class="mo">/</span></span></span><span id="MathJax-Span-890" class="msup"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-891" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.61em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-892" class="mo">′</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.081em; border-left: 0px solid; width: 0px; height: 1.134em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>L</mi><mo>′</mo></msup></math></span></span><script id="MathJax-Element-94" type="math/tex">L / L'</script></span>, with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>L</mi><mo>&#x2032;</mo></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-95-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.203em; display: inline-block;" role="math" id="MathJax-Span-893" class="math"><span style="display: inline-block; position: relative; width: 0.913em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.545em, 1000.91em, 2.596em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-894" class="mrow"><span id="MathJax-Span-895" class="msup"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-896" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.61em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-897" class="mo">′</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.116em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>L</mi><mo>′</mo></msup></math></span></span><script id="MathJax-Element-95" type="math/tex">L'</script></span> an abelian number field contained in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-96-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.766em; display: inline-block;" role="math" id="MathJax-Span-898" class="math"><span style="display: inline-block; position: relative; width: 0.577em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.58em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-899" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-900" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math></span></span><script id="MathJax-Element-96" type="math/tex">L</script></span>, is the subgroup in this group that stabilizes <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>L</mi><mo>&#x2032;</mo></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-97-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.203em; display: inline-block;" role="math" id="MathJax-Span-901" class="math"><span style="display: inline-block; position: relative; width: 0.913em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.545em, 1000.91em, 2.596em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-902" class="mrow"><span id="MathJax-Span-903" class="msup"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-904" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.61em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-905" class="mo">′</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.116em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>L</mi><mo>′</mo></msup></math></span></span><script id="MathJax-Element-97" type="math/tex">L'</script></span>. These groups are easily described in terms of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-98-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.328em; display: inline-block;" role="math" id="MathJax-Span-906" class="math"><span style="display: inline-block; position: relative; width: 3.317em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.804em, 1003.32em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-907" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-908" class="mo">(</span><span id="MathJax-Span-909" class="texatom"><span id="MathJax-Span-910" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-911" class="mo">ℤ</span></span></span><span id="MathJax-Span-912" class="texatom"><span id="MathJax-Span-913" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-914" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-915" class="mi">n</span><span id="MathJax-Span-916" class="texatom"><span id="MathJax-Span-917" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-918" class="mo">ℤ</span></span></span><span id="MathJax-Span-919" class="msubsup"><span style="display: inline-block; position: relative; width: 0.778em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-920" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span id="MathJax-Span-921" class="texatom"><span id="MathJax-Span-922" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-923" class="mo">∗</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.26em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo></mrow></msup></math></span></span><script id="MathJax-Element-98" type="math/tex">(ℤ / n ℤ)^{*}</script></span>. Generators of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2217;</mo></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-99-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.328em; display: inline-block;" role="math" id="MathJax-Span-924" class="math"><span style="display: inline-block; position: relative; width: 3.317em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.804em, 1003.32em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-925" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-926" class="mo">(</span><span id="MathJax-Span-927" class="texatom"><span id="MathJax-Span-928" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-929" class="mo">ℤ</span></span></span><span id="MathJax-Span-930" class="texatom"><span id="MathJax-Span-931" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-932" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-933" class="mi">n</span><span id="MathJax-Span-934" class="texatom"><span id="MathJax-Span-935" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-936" class="mo">ℤ</span></span></span><span id="MathJax-Span-937" class="msubsup"><span style="display: inline-block; position: relative; width: 0.778em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-938" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span id="MathJax-Span-939" class="texatom"><span id="MathJax-Span-940" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-941" class="mo">∗</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.26em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>∗</mo></mrow></msup></math></span></span><script id="MathJax-Element-99" type="math/tex">(ℤ / n ℤ)^{*}</script></span> can be computed using <code class="func">GeneratorsPrimeResidues</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap15_mj.html#X7D191CF67E5018BE"><span class="RefLink">15.2-4</span></a>).</p>
-
-<p>In <strong class="pkg">GAP</strong>, a field extension <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>L</mi><mo>&#x2032;</mo></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-100-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.328em; display: inline-block;" role="math" id="MathJax-Span-942" class="math"><span style="display: inline-block; position: relative; width: 1.779em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.738em, 1001.78em, 2.802em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-943" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-944" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span id="MathJax-Span-945" class="texatom"><span id="MathJax-Span-946" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-947" class="mo">/</span></span></span><span id="MathJax-Span-948" class="msup"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-949" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.61em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-950" class="mo">′</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.081em; border-left: 0px solid; width: 0px; height: 1.134em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>L</mi><mo>′</mo></msup></math></span></span><script id="MathJax-Element-100" type="math/tex">L / L'</script></span> is given by the field object <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-101-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.766em; display: inline-block;" role="math" id="MathJax-Span-951" class="math"><span style="display: inline-block; position: relative; width: 0.577em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.58em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-952" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-953" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math></span></span><script id="MathJax-Element-101" type="math/tex">L</script></span> with <code class="func">LeftActingDomain</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap57_mj.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>) value <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>L</mi><mo>&#x2032;</mo></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-102-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.203em; display: inline-block;" role="math" id="MathJax-Span-954" class="math"><span style="display: inline-block; position: relative; width: 0.913em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.545em, 1000.91em, 2.596em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-955" class="mrow"><span id="MathJax-Span-956" class="msup"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-957" class="mi">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.61em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-958" class="mo">′</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.116em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>L</mi><mo>′</mo></msup></math></span></span><script id="MathJax-Element-102" type="math/tex">L'</script></span> (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(15);</span>
-CF(15)
-<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GaloisGroup( f );</span>
-<group with 2 generators>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Size( g ); IsCyclic( g ); IsAbelian( g );</span>
-8
-false
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">Action( g, NormalBase( f ), OnPoints );</span>
-Group([ (1,6)(2,4)(3,8)(5,7), (1,4,3,7)(2,8,5,6) ])
-</pre></div>
-
-<p>The following example shows Galois groups of a cyclotomic field and of a proper subfield that is not a cyclotomic field.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">gens1:= GeneratorsOfGroup( GaloisGroup( CF(5) ) );</span>
-[ ANFAutomorphism( CF(5), 2 ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">gens2:= GeneratorsOfGroup( GaloisGroup( Field( Sqrt(5) ) ) );</span>
-[ ANFAutomorphism( NF(5,[ 1, 4 ]), 2 ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Order( gens1[1] ); Order( gens2[1] );</span>
-4
-2
-<span class="GAPprompt">gap></span> <span class="GAPinput">Sqrt(5)^gens1[1] = Sqrt(5)^gens2[1];</span>
-true
-</pre></div>
-
-<p>The following example shows the Galois group of a cyclotomic field over a non-cyclotomic field.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">g:= GaloisGroup( AsField( Field( [ Sqrt(5) ] ), CF(5) ) );</span>
-<group with 1 generators>
-<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( g );</span>
-[ ANFAutomorphism( AsField( NF(5,[ 1, 4 ]), CF(5) ), 4 ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">x:= last[1];; x^2;</span>
-IdentityMapping( AsField( NF(5,[ 1, 4 ]), CF(5) ) )
-</pre></div>
-
-<p><a id="X8643D4B47A827D9D" name="X8643D4B47A827D9D"></a></p>
-
-<h5>60.4-2 ANFAutomorphism</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ANFAutomorphism</code>( <var class="Arg">F</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">F</var> be an abelian number field and <var class="Arg">k</var> be an integer that is coprime to the conductor (see <code class="func">Conductor</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>)) of <var class="Arg">F</var>. Then <code class="func">ANFAutomorphism</code> returns the automorphism of <var class="Arg">F</var> that is defined as the linear extension of the map that raises each root of unity in <var class="Arg">F</var> to its <var class="Arg">k</var>-th power.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(25);</span>
-CF(25)
-<span class="GAPprompt">gap></span> <span class="GAPinput">alpha:= ANFAutomorphism( f, 2 );</span>
-ANFAutomorphism( CF(25), 2 )
-<span class="GAPprompt">gap></span> <span class="GAPinput">alpha^2;</span>
-ANFAutomorphism( CF(25), 4 )
-<span class="GAPprompt">gap></span> <span class="GAPinput">Order( alpha );</span>
-20
-<span class="GAPprompt">gap></span> <span class="GAPinput">E(5)^alpha;</span>
-E(5)^2
-</pre></div>
-
-<p><a id="X85E9E90D7FE877CC" name="X85E9E90D7FE877CC"></a></p>
-
-<h4>60.5 <span class="Heading">Gaussians</span></h4>
-
-<p><a id="X80BD5EAB879F096E" name="X80BD5EAB879F096E"></a></p>
-
-<h5>60.5-1 GaussianIntegers</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ GaussianIntegers</code></td><td class="tdright">( global variable )</td></tr></tbody></table></div>
-<p><code class="func">GaussianIntegers</code> is the ring <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow><mo stretchy="false">[</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msqrt><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-103-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.391em; display: inline-block;" role="math" id="MathJax-Span-959" class="math"><span style="display: inline-block; position: relative; width: 3.365em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.67em, 1003.28em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-960" class="mrow"><span id="MathJax-Span-961" class="texatom"><span id="MathJax-Span-962" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-963" class="mo">ℤ</span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-964" class="mo">[</span><span id="MathJax-Span-965" class="msqrt"><span style="display: inline-block; position: relative; width: 1.947em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1001.08em, 4.135em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-966" class="mrow"><span id="MathJax-Span-967" class="texatom"><span id="MathJax-Span-968" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-969" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-970" class="mn">1</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1001.21em, 3.365em, -1000em); top: -4em; left: 0.737em;"><span style="display: inline-block; position: relative; width: 1.21em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0.71em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.99em; left: 0.343em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.877em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-971" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.407em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow><mo stretchy="false">[</mo><msqrt><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msqrt><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-103" type="math/tex">ℤ[\sqrt{{-1}}]</script></span> of Gaussian integers. This is a subring of the cyclotomic field <code class="func">GaussianRationals</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X82F53C65802FF551"><span class="RefLink">60.1-3</span></a>).</p>
-
-<p><a id="X7BFD33D27BFB7C5A" name="X7BFD33D27BFB7C5A"></a></p>
-
-<h5>60.5-2 IsGaussianIntegers</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsGaussianIntegers</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<p>is the defining category for the domain <code class="func">GaussianIntegers</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html#X80BD5EAB879F096E"><span class="RefLink">60.5-1</span></a>).</p>
-
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href="http://www.gap-system.org/Manuals/doc/ref/chap77_mj.html">77</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap78_mj.html">78</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap79_mj.html">79</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap80_mj.html">80</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap81_mj.html">81</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap82_mj.html">82</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap83_mj.html">83</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap84_mj.html">84</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap85_mj.html">85</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap86_mj.html">86</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap87_mj.html">87</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html">Bib</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chapInd_mj.html">Ind</a> </div>
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-<div class="ChapSects"><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7B7A9EE881E01C10">71 <span class="Heading">Character Tables</span></a>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D9C08FE81AE1102">71.1 <span class="Heading">Some Remarks about Character Theory in <strong class="pkg">GAP</strong></span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7F8AB7CB7A46002F">71.2 <span class="Heading">History of Character Theory Stuff in GAP</span></a>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8701174D86B586AF">71.3 <span class="Heading">Creating Character Tables</span></a>
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-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FCA7A7A822BDA33">71.3-1 <span class="Heading">CharacterTable</span></a>
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-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8476B25A79D7A7FC">71.3-2 <span class="Heading">BrauerTable</span></a>
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-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X85DB8AE7786A2DB5">71.3-3 CharacterTableRegular</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7DBEF4BF87F10CD6">71.3-4 SupportedCharacterTableInfo</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8195BC057B1DFAD5">71.3-5 ConvertToCharacterTable</a></span>
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-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X789FAC077AEF088A">71.4 <span class="Heading">Character Table Categories</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X82FF82C87CF82ADF">71.4-1 IsNearlyCharacterTable</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C6F3D947E5188D1">71.4-2 InfoCharacterTable</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FA867637EBB30F9">71.4-3 NearlyCharacterTablesFamily</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X829C4B6E83998F40">71.5 <span class="Heading">Conventions for Character Tables</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X793E0EBF84B07313">71.6 <span class="Heading">The Interface between Character Tables and Groups</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FF4826A82B667AF">71.6-1 UnderlyingGroup</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X849A38F887F6EC86">71.6-2 ConjugacyClasses</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X84DC12AA804C8085">71.6-3 IdentificationOfConjugacyClasses</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8788C6C7829C1ADE">71.6-4 CharacterTableWithStoredGroup</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X790019E87CFDDB98">71.6-5 CompatibleConjugacyClasses</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CADCBC9824CB624">71.7 <span class="Heading">Operators for Character Tables</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7F9D58208241D35E">71.8 <span class="Heading">Attributes and Properties for Groups and Character Tables</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X81FEFF768134481A">71.8-1 <span class="Heading">CharacterDegrees</span></a>
-</span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492">71.8-2 <span class="Heading">Irr</span></a>
-</span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8549899A7DE206BA">71.8-3 <span class="Heading">LinearCharacters</span></a>
-</span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8011EEB684848039">71.8-4 <span class="Heading">OrdinaryCharacterTable</span></a>
-</span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X81EFD9FE804AC6EE">71.8-5 <span class="Heading">Group Operations Applicable to Character Tables</span></a>
-</span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7995A2AD83BC58A0">71.9 <span class="Heading">Attributes and Properties only for Character Tables</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86F455DA7A9C30EE">71.9-1 OrdersClassRepresentatives</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CF7907F790A5DE6">71.9-2 SizesCentralizers</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D9D2A45879A6A62">71.9-3 SizesConjugacyClasses</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C2753DE8094F4BA">71.9-4 AutomorphismsOfTable</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7F58A82F7D88000A">71.9-5 <span class="Heading">UnderlyingCharacteristic</span></a>
-</span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X804CFD597C795801">71.9-6 <span class="Heading">Class Names and Character Names</span></a>
-</span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8333E8038308947E">71.9-7 <span class="Heading">Class Parameters and Character Parameters</span></a>
-</span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79C40EE97890202F">71.9-8 Identifier</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7932C35180C80953">71.9-9 InfoText</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7919E2897BE8234A">71.9-10 InverseClasses</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87FF547981456932">71.9-11 RealClasses</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7ABB007C799F7C49">71.9-12 ClassOrbit</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7F863B15804E0835">71.9-13 ClassRoots</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79CEBC3C7E0E63DF">71.10 <span class="Heading">Normal Subgroups Represented by Lists of Class Positions</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X850C7D947B3DBFA2">71.10-1 ClassPositionsOfNormalSubgroups</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8491DA0981D6F264">71.10-2 ClassPositionsOfAgemo</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7A6B1F8A84A495DC">71.10-3 ClassPositionsOfCentre</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D53F60785AB22B1">71.10-4 ClassPositionsOfDirectProductDecompositions</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79EE7BE17BD343D5">71.10-5 ClassPositionsOfDerivedSubgroup</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86ABB2E179D7F6E1">71.10-6 ClassPositionsOfElementaryAbelianSeries</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D2A55A584F955DB">71.10-7 ClassPositionsOfFittingSubgroup</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79AEFC4384769B72">71.10-8 ClassPositionsOfLowerCentralSeries</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86065D217A36CD9B">71.10-9 ClassPositionsOfUpperCentralSeries</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8392DD5B813250A4">71.10-10 ClassPositionsOfSupersolvableResiduum</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7BBE7EBA7A64A6B0">71.10-11 ClassPositionsOfPCore</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FCF905D7FD7CC40">71.10-12 ClassPositionsOfNormalClosure</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8733F0EA801785D4">71.11 <span class="Heading">Operations Concerning Blocks</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7ACB9306804F4E3F">71.11-1 PrimeBlocks</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E80E35985275F35">71.11-2 SameBlock</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FF4CE4A7A272F88">71.11-3 BlocksInfo</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X84701640811D2345">71.11-4 DecompositionMatrix</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X83EC921380AF9B3B">71.11-5 LaTeXStringDecompositionMatrix</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873211618402ACF7">71.12 <span class="Heading">Other Operations for Character Tables</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8441983C845F2176">71.12-1 Index</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8123650E817926FC">71.12-2 IsInternallyConsistent</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7A0CBD1884276882">71.12-3 IsPSolvableCharacterTable</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X82F523E8784B3752">71.12-4 IsClassFusionOfNormalSubgroup</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FD3D3047DE6381E">71.12-5 Indicator</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X83AE05BF8085B3C8">71.12-6 NrPolyhedralSubgroups</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E2EA9FE7D3062D3">71.12-7 ClassMultiplicationCoefficient</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7A19F56C7FD5EFC7">71.12-8 ClassStructureCharTable</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X809E67E57D4933B3">71.12-9 MatClassMultCoeffsCharTable</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C1941F17BE9FC21">71.13 <span class="Heading">Printing Character Tables</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D45224B86D802E5">71.13-1 ViewObj</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X836554207C678D41">71.13-2 PrintObj</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7B41F36478C47364">71.13-3 Display</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X85E883A87A190AA4">71.13-4 DisplayOptions</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79EC9603833AA2AB">71.13-5 PrintCharacterTable</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79BC08C6846718D9">71.14 <span class="Heading">Computing the Irreducible Characters of a Group</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7ED39DB680BFEA96">71.14-1 IrrDixonSchneider</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E81BCD686561DF0">71.14-2 IrrConlon</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7BF15729839203FC">71.14-3 IrrBaumClausen</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7F29C5447B5DC102">71.14-4 IrreducibleRepresentations</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8493ED7A86FFCB8A">71.14-5 IrreducibleRepresentationsDixon</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E51AACD79CE0BC8">71.15 <span class="Heading">Representations Given by Modules</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87E82F8085745523">71.15-1 IrreducibleModules</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D0BD5337D1C069B">71.15-2 AbsolutelyIrreducibleModules</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7EB88B2E87AF5556">71.15-3 RegularModule</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86CDA4007A5EF704">71.16 <span class="Heading">The Dixon-Schneider Algorithm</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C083207868066C1">71.17 <span class="Heading">Advanced Methods for Dixon-Schneider Calculations</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C398F2680C8616B">71.17-1 DixonRecord</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E33C03E7BDDC9B0">71.17-2 DixonInit</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X868476037907918F">71.17-3 DixontinI</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87ABE0B081DAD476">71.17-4 DixonSplit</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7BFD4C1A821731FB">71.17-5 BestSplittingMatrix</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C85B56C80BFA2E3">71.17-6 DxIncludeIrreducibles</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87A5B5C77F7F348E">71.17-7 SplitCharacters</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8089009E7EA85BC8">71.17-8 IsDxLargeGroup</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C1153637E7D2133">71.18 <span class="Heading">Components of a Dixon Record</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X782B5E37848786BC">71.19 <span class="Heading">An Example of Advanced Dixon-Schneider Calculations</span></a>
-</span>
-</div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C38C5067941D496">71.20 <span class="Heading">Constructing Character Tables from Others</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7BE1572D7BBC6AC8">71.20-1 CharacterTableDirectProduct</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C97CF727FBDFCAB">71.20-2 FactorsOfDirectProduct</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C3A4E5283B240BE">71.20-3 CharacterTableFactorGroup</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X85BE46F784B83938">71.20-4 CharacterTableIsoclinic</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79B75C8582426BC5">71.20-5 CharacterTableWreathSymmetric</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X816FCD5A805F9FE8">71.21 <span class="Heading">Sorted Character Tables</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D9C4A7F8086F671">71.21-1 CharacterTableWithSortedCharacters</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87E3CF317D8E4EC7">71.21-2 SortedCharacters</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E3DE0A47E62BE6B">71.21-3 CharacterTableWithSortedClasses</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X82DCAAA882416E24">71.21-4 SortedCharacterTable</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8099FEDC7DE03AEE">71.21-5 ClassPermutation</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7B0A669484470D09">71.22 <span class="Heading">Automorphisms and Equivalence of Character Tables</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X84353BB884AF0365">71.22-1 MatrixAutomorphisms</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8082DD827C673138">71.22-2 TableAutomorphisms</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D721E3D7AA319F5">71.22-3 TransformingPermutations</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X849731AA7EC9FA73">71.22-4 TransformingPermutationsCharacterTables</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8117D940835B0B47">71.22-5 FamiliesOfRows</a></span>
-</div></div>
-<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X81272CEE79F13E7B">71.23 <span class="Heading">Storing Normal Subgroup Information</span></a>
-</span>
-<div class="ContSSBlock">
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E66174C7C7A8C0C">71.23-1 NormalSubgroupClassesInfo</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C2A87E085111090">71.23-2 ClassPositionsOfNormalSubgroup</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87E7391F7F92377C">71.23-3 NormalSubgroupClasses</a></span>
-<span class="ContSS"><br><span class="nocss"> </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79D451F0808EB252">71.23-4 FactorGroupNormalSubgroupClasses</a></span>
-</div></div>
-</div>
-
-<h3>71 <span class="Heading">Character Tables</span></h3>
-
-<p>This chapter describes operations for <em>character tables of finite groups</em>.</p>
-
-<p>Operations for <em>characters</em> (or, more general, <em>class functions</em>) are described in Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X7C91D0D17850E564"><span class="RefLink">72</span></a>.</p>
-
-<p>For a description of the <strong class="pkg">GAP</strong> Library of Character Tables, see the separate manual for the <strong class="pkg">GAP</strong> package <strong class="pkg">CTblLib</strong>.</p>
-
-<p>Several examples in this chapter require the <strong class="pkg">GAP</strong> Character Table Library to be available. If it is not yet loaded then we load it now.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "ctbllib" );</span>
-true
-</pre></div>
-
-<p><a id="X7D9C08FE81AE1102" name="X7D9C08FE81AE1102"></a></p>
-
-<h4>71.1 <span class="Heading">Some Remarks about Character Theory in <strong class="pkg">GAP</strong></span></h4>
-
-<p>It seems to be necessary to state some basic facts –and maybe
-warnings– at the beginning of the character theory package. This holds
-for people who are familiar with character theory because there is no
-global reference on computational character theory, although there are
-many papers on this topic, such as <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBNPP84">[NPP84]</a> or <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBLP91">[LP91]</a>. It holds, however, also for people who are familiar with <strong class="pkg">GAP</strong> because the general concept of domains (see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap12_mj.html#X7BAF69417BB925F6"><span class="RefLink">12.4</span></a>) plays no important role here –we will justify this later in this section.</p>
-
-<p>Intuitively, <em>characters</em> (or more generally, <em>class functions</em>) of a finite group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-1-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-1" type="math/tex">G</script></span> can be thought of as certain mappings defined on <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-2-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-4" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-5" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-6" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-2" type="math/tex">G</script></span>, with values in the complex number field; the set of all characters of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-3-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-7" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-8" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-9" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-3" type="math/tex">G</script></span> forms a semiring, with both addition and multiplication defined pointwise, which is naturally embedded into the ring of <em>generalized</em> (or <em>virtual</em>) <em>characters</em> in the natural way. A <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2124;</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-4-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-10" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.838em, 1000.67em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-11" class="mrow"><span id="MathJax-Span-12" class="texatom"><span id="MathJax-Span-13" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-14" class="mo">ℤ</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.986em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo>ℤ</mo></mrow></math></span></span><script id="MathJax-Element-4" type="math/tex">ℤ</script></span>-basis of this ring, and also a vector space basis of the complex vector space of class functions of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-5-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-15" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-16" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-17" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-5" type="math/tex">G</script></span>, is given by the irreducible characters of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-6-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-18" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-19" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-20" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-6" type="math/tex">G</script></span>.</p>
-
-<p>At this stage one could ask where there is a problem, since all these algebraic structures are supported by <strong class="pkg">GAP</strong>. But in practice, these structures are of minor importance, compared to individual characters and the <em>character tables</em> themselves (which are not domains in the sense of <strong class="pkg">GAP</strong>).</p>
-
-<p>For computations with characters of a finite group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-7-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-21" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-22" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-23" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-7" type="math/tex">G</script></span> with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-8-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-24" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-25" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-26" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-8" type="math/tex">n</script></span>
- conjugacy classes, say, we fix an ordering of the classes, and then
-identify each class with its position according to this ordering. Each
-character of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-9-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-27" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-28" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-29" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-9" type="math/tex">G</script></span> can be represented by a list of length <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-10-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-30" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-31" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-32" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-10" type="math/tex">n</script></span> in which the character value for elements of the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-11-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-33" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-34" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-35" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-11" type="math/tex">i</script></span>-th class is stored at the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-12-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-36" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-37" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-38" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-12" type="math/tex">i</script></span>-th position. Note that we need not know the conjugacy classes of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-13-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-39" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-40" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-41" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-13" type="math/tex">G</script></span> physically, even our knowledge of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-14-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-42" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-43" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-44" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-14" type="math/tex">G</script></span> may be implicit in the sense that, e.g., we know how many classes of involutions <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-15-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-45" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-46" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-47" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-15" type="math/tex">G</script></span> has, and which length these classes have, but we never have seen an element of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-16-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-48" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-49" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-50" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-16" type="math/tex">G</script></span>, or a presentation or representation of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-17-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-51" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-52" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-53" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-17" type="math/tex">G</script></span>. This allows us to work with the character tables of very large groups, e.g., of the so-called monster, where <strong class="pkg">GAP</strong> has (currently) no chance to deal with the group.</p>
-
-<p>As a consequence, also other information involving characters is
-given implicitly. For example, we can talk about the kernel of a
-character not as a group but as a list of classes (more exactly: a list
-of their positions according to the chosen ordering of classes) forming
-this kernel; we can deduce the group order, the contained cyclic
-subgroups and so on, but we do not get the group itself.</p>
-
-<p>So typical calculations with characters involve loops over lists of
-character values. For example, the scalar product of two characters <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C7;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-18-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-54" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.53em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-55" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-56" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 0.967em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>χ</mi></math></span></span><script id="MathJax-Element-18" type="math/tex">\chi</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C8;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-19-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-57" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.67em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-58" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-59" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ψ</mi></math></span></span><script id="MathJax-Element-19" type="math/tex">\psi</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-20-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-60" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-61" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-62" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-20" type="math/tex">G</script></span> given by</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"></span><div style="text-align: center;" class="MathJax_Display"><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">[</mo><mi>&#x03C7;</mi><mo>,</mo><mi>&#x03C8;</mi><mo stretchy="false">]</mo><mo>=</mo><mrow><mo>(</mo><munder><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>g</mi><mo>&#x2208;</mo><mi>G</mi></mrow></mrow></munder><mi>&#x03C7;</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mi>&#x03C8;</mi><mo stretchy="false">(</mo><msup><mi>g</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></msup><mo stretchy="false">)</mo><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-21-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 16.578em; display: inline-block;" role="math" id="MathJax-Span-63" class="math"><span style="display: inline-block; position: relative; width: 12.74em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.251em, 1012.67em, 5.723em, -1000em); top: -4.183em; left: 0em;"><span id="MathJax-Span-64" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-65" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-66" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-67" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-68" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-69" class="mo">]</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-70" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-71" class="mrow"><span style="vertical-align: -0.778em;" id="MathJax-Span-72" class="mo"><span style="font-family: STIXSizeFourSym;">(</span></span><span id="MathJax-Span-73" class="munderover"><span style="display: inline-block; position: relative; width: 1.348em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0.028em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-74" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.375em, 1001.35em, 4.38em, -1000em); top: -2.84em; left: 0em;"><span id="MathJax-Span-75" class="texatom"><span id="MathJax-Span-76" class="mrow"><span id="MathJax-Span-77" class="texatom"><span id="MathJax-Span-78" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-79" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-80" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-81" class="mi">G</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-82" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-83" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-84" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-85" class="mo">)</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-86" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-87" class="mo">(</span><span id="MathJax-Span-88" class="msubsup"><span style="display: inline-block; position: relative; width: 1.413em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-89" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.5em;"><span id="MathJax-Span-90" class="texatom"><span id="MathJax-Span-91" class="mrow"><span id="MathJax-Span-92" class="texatom"><span id="MathJax-Span-93" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-94" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-95" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-96" class="mo">)</span><span style="vertical-align: -0.778em;" id="MathJax-Span-97" class="mo"><span style="font-family: STIXSizeFourSym;">)</span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-98" class="texatom"><span id="MathJax-Span-99" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-100" class="mo">/</span></span></span><span id="MathJax-Span-101" class="texatom"><span id="MathJax-Span-102" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-103" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-104" class="mi">G</span><span id="MathJax-Span-105" class="texatom"><span id="MathJax-Span-106" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-107" class="mo">|</span></span></span></span><span style="display: inline-block; width: 0px; height: 4.183em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.877em; border-left: 0px solid; width: 0px; height: 4.263em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML MJX_Assistive_MathML_Block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">[</mo><mi>χ</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">]</mo><mo>=</mo><mrow><mo>(</mo><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></mrow></munder><mi>χ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">(</mo><msup><mi>g</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></mrow></msup><mo stretchy="false">)</mo><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></math></span></span></div><script id="MathJax-Element-21" type="math/tex; mode=display">
-[ \chi, \psi ] =
-\left( \sum_{{g \in G}} \chi(g) \psi(g^{{-1}}) \right) / |G|
-</script></p>
-
-<p>can be written as</p>
-
-
-<div class="example"><pre>Sum( [ 1 .. n ], i -> SizesConjugacyClasses( t )[i] * chi[i]
- * ComplexConjugate( psi[i] ) ) / Size( t );
-</pre></div>
-
-<p>where <code class="code">t</code> is the character table of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-22-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-108" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-109" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-110" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-22" type="math/tex">G</script></span>, and <code class="code">chi</code>, <code class="code">psi</code> are the lists of values of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C7;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-23-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-111" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.53em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-112" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-113" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 0.967em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>χ</mi></math></span></span><script id="MathJax-Element-23" type="math/tex">\chi</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C8;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-24-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-114" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.67em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-115" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-116" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ψ</mi></math></span></span><script id="MathJax-Element-24" type="math/tex">\psi</script></span>, respectively.</p>
-
-<p>It is one of the advantages of character theory that after one has
-translated a problem concerning groups into a problem concerning only
-characters, the necessary calculations are mostly simple. For example,
-one can often prove that a group is a Galois group over the rationals
-using calculations with structure constants that can be computed from
-the character table, and information about (the character tables of)
-maximal subgroups. When one deals with such questions, the translation
-back to groups is just an interpretation by the user, it does not take
-place in <strong class="pkg">GAP</strong>.</p>
-
-<p><strong class="pkg">GAP</strong> uses character <em>tables</em> to store information such as class lengths, element orders, the irreducible characters of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-25-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-117" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-118" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-119" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-25" type="math/tex">G</script></span> etc. in a consistent way; in the example above, we have seen that <code class="func">SizesConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D9D2A45879A6A62"><span class="RefLink">71.9-3</span></a>)
- returns the list of class lengths of its argument. Note that the values
- of these attributes rely on the chosen ordering of conjugacy classes, a
- character table is not determined by something similar to generators of
- groups or rings in <strong class="pkg">GAP</strong> where knowledge could in principle be recovered from the generators but is stored mainly for the sake of efficiency.</p>
-
-<p>Note that the character table of a group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-26-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-120" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-121" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-122" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-26" type="math/tex">G</script></span> in <strong class="pkg">GAP</strong> must <em>not</em> be mixed up with the list of complex irreducible characters of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-27-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-123" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-124" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-125" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-27" type="math/tex">G</script></span>. The irreducible characters are stored in a character table via the attribute <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>).</p>
-
-<p>Two further important instances of information that depends on the ordering of conjugacy classes are <em>power maps</em> and <em>fusion maps</em>. Both are represented as lists of integers in <strong class="pkg">GAP</strong>. The <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-28-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-126" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.43em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-127" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-128" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script id="MathJax-Element-28" type="math/tex">k</script></span>-th power map maps each class to the class of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-29-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-129" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.43em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-130" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-131" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script id="MathJax-Element-29" type="math/tex">k</script></span>-th
- powers of its elements, the corresponding list contains at each
-position the position of the image. A class fusion map between the
-classes of a subgroup <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-30-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-132" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.77em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-133" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-134" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math></span></span><script id="MathJax-Element-30" type="math/tex">H</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-31-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-135" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-136" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-137" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-31" type="math/tex">G</script></span> and the classes of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-32-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-138" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-139" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-140" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-32" type="math/tex">G</script></span> maps each class <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-33-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-141" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.41em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-142" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-143" class="mi">c</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.713em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script id="MathJax-Element-33" type="math/tex">c</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-34-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-144" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.77em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-145" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-146" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math></span></span><script id="MathJax-Element-34" type="math/tex">H</script></span> to that class of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-35-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-147" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-148" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-149" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-35" type="math/tex">G</script></span> that contains <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-36-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-150" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.41em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-151" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-152" class="mi">c</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.713em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script id="MathJax-Element-36" type="math/tex">c</script></span>, the corresponding list contains again the positions of image classes; if we know only the character tables of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-37-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-153" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.77em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-154" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-155" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math></span></span><script id="MathJax-Element-37" type="math/tex">H</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-38-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-156" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-157" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-158" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-38" type="math/tex">G</script></span> but not the groups themselves, this means with respect to a fixed embedding of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-39-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-159" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.77em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-160" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-161" class="mi">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.047em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math></span></span><script id="MathJax-Element-39" type="math/tex">H</script></span> into <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-40-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-162" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-163" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-164" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-40" type="math/tex">G</script></span>. More about power maps and fusion maps can be found in Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X7DF1ACDE7E9C6294"><span class="RefLink">73</span></a>.</p>
-
-<p>So class functions, power maps, and fusion maps are represented by lists in <strong class="pkg">GAP</strong>.
- If they are plain lists then they are regarded as class functions
-etc. of an appropriate character table when they are passed to <strong class="pkg">GAP</strong>
- functions that expect class functions etc. For example, a list with all
- entries equal to 1 is regarded as the trivial character if it is passed
- to a function that expects a character. Note that this approach
-requires the character table as an argument for such a function.</p>
-
-<p>One can construct class function objects that store their underlying
-character table and other attribute values (see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X7C91D0D17850E564"><span class="RefLink">72</span></a>).
- This allows one to omit the character table argument in many functions,
- and it allows one to use infix operations for tensoring or inducing
-class functions.</p>
-
-<p><a id="X7F8AB7CB7A46002F" name="X7F8AB7CB7A46002F"></a></p>
-
-<h4>71.2 <span class="Heading">History of Character Theory Stuff in GAP</span></h4>
-
-<p><strong class="pkg">GAP</strong> provides functions for dealing with group characters since the version <strong class="pkg">GAP</strong> 3.1, which was released in March 1992. The reason for adding this branch of mathematics to the topics of <strong class="pkg">GAP</strong> was (apart from the usefulness of character theoretic computations in general) the insight that <strong class="pkg">GAP</strong>
- provides an ideal environment for developing the algorithms needed. In
-particular, it had been decided at Lehrstuhl D für Mathematik that the <strong class="pkg">CAS</strong> system (a standalone Fortran program together with a database of character tables, see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBNPP84">[NPP84]</a>) should not be developed further and the functionality of <strong class="pkg">CAS</strong> should be made available in <strong class="pkg">GAP</strong>. The background was that extending <strong class="pkg">CAS</strong> (by new Fortran code) had turned out to be much less flexible than writing analogous <strong class="pkg">GAP</strong> library code.</p>
-
-<p>For integrating the existing character theory algorithms, <strong class="pkg">GAP</strong>'s
- memory management and long integer arithmetic were useful as well as
-the list handling –it is an important feature of character theoretic
-methods that questions about groups are translated into manipulations of
- lists; on the other hand, the datatype of cyclotomics (see
-Chapter <code class="func">Cyclotomics</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X863D1E017BC9EB7F"><span class="RefLink">18.1-2</span></a>)) was added to the <strong class="pkg">GAP</strong> kernel because of the character theory algorithms. For developing further code, also other areas of <strong class="pkg">GAP</strong>'s library became interesting, such as permutation groups, finite fields, and polynomials.</p>
-
-<p>The development of character theory code for <strong class="pkg">GAP</strong>
- has been supported by several DFG grants, in particular the project
-"Representation Theory of Finite Groups and Finite Dimensional Algebras"
- (until 1991), and the Schwerpunkt "Algorithmische Zahlentheorie und
-Algebra" (from 1991 until 1997). Besides that, several Diploma theses at
- Lehrstuhl D were concerned with the development and/or implementation
-of algorithms dealing with characters in <strong class="pkg">GAP</strong>.</p>
-
-<p>The major contributions can be listed as follows.</p>
-
-
-<ul>
-<li><p>The arithmetic for the cyclotomics data type, following <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBZum89">[Zum89]</a>,
- was first implemented by Marco van Meegen; an alternative approach was
-studied in the diploma thesis of Michael Scherner (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBScherner92">[Sch92]</a>) but was not efficient enough; later Martin Schönert replaced the implementation by a better one.</p>
-
-</li>
-<li><p>The basic routines for characters and character tables were written by Thomas Breuer and Götz Pfeiffer.</p>
-
-</li>
-<li><p>The lattice related functions, such as <code class="func">LLL</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X85B360C085B360C0"><span class="RefLink">72.10-4</span></a>), <code class="func">OrthogonalEmbeddings</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap25_mj.html#X842280C2808FF05D"><span class="RefLink">25.6-1</span></a>), and <code class="func">DnLattice</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X85D510DC873A99B4"><span class="RefLink">72.10-8</span></a>), were implemented by Ansgar Kaup (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBKaup92">[Kau92]</a>).</p>
-
-</li>
-<li><p>Functions for computing possible class fusions, possible power
-maps, and table automorphisms were written by Thomas Breuer (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBBre91">[Bre91]</a>).</p>
-
-</li>
-<li><p>Functions for computing possible permutation characters were written by Thomas Breuer (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBBre91">[Bre91]</a>) and Götz Pfeiffer (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBPfe91">[Pfe91]</a>).</p>
-
-</li>
-<li><p>Functions for computing character tables from groups were written by Alexander Hulpke (Dixon-Schneider algorithm, see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBHulpke93">[Hul93]</a>) and Hans Ulrich Besche (Baum algorithm and Conlon algorithm, see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBBesche92">[Bes92]</a>).</p>
-
-</li>
-<li><p>Functions for dealing with Clifford matrices were written by Ute Schiffer (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBSchiffer94">[Sch94]</a>).</p>
-
-</li>
-<li><p>Functions for monomiality questions were written by Thomas Breuer and Erzsébet Horváth.</p>
-
-</li>
-</ul>
-<p>Since then, the code has been maintained and extended further by
-Alexander Hulpke (code related to his implementation of the
-Dixon-Schneider algorithm) and Thomas Breuer.</p>
-
-<p>Currently <strong class="pkg">GAP</strong> does not provide special functionality for computing Brauer character tables, but there is an interface to the <strong class="pkg">MOC</strong> system (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBHJLP92">[HJLP]</a>), and the <strong class="pkg">GAP</strong> Character Table Library contains many known Brauer character tables.</p>
-
-<p><a id="X8701174D86B586AF" name="X8701174D86B586AF"></a></p>
-
-<h4>71.3 <span class="Heading">Creating Character Tables</span></h4>
-
-<p>There are in general five different ways to get a character table in <strong class="pkg">GAP</strong>. You can</p>
-
-<ol>
-<li><p>compute the table from a group,</p>
-
-</li>
-<li><p>read a file that contains the table data,</p>
-
-</li>
-<li><p>construct the table using generic formulae,</p>
-
-</li>
-<li><p>derive it from known character tables, or</p>
-
-</li>
-<li><p>combine partial information about conjugacy classes, power maps
-of the group in question, and about (character tables of) some subgroups
- and supergroups.</p>
-
-</li>
-</ol>
-<p>In 1., the computation of the irreducible characters is the hardest
-part; the different algorithms available for this are described in <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79BC08C6846718D9"><span class="RefLink">71.14</span></a>. Possibility 2. is used for the character tables in the <strong class="pkg">GAP</strong> Character Table Library, see the manual of this library. Generic character tables –as addressed by 3.– are described in <a href="http://www.gap-system.org/Manuals/pkg/ctbllib/doc/chap4.html#X81E3F9A384365282"><span class="RefLink">CTblLib: Generic Character Tables</span></a>. Several occurrences of 4. are described in <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C38C5067941D496"><span class="RefLink">71.20</span></a>. The last of the above possibilities <em>is currently not supported and will be described in a chapter of its own when it becomes available</em>.</p>
-
-<p>The operation <code class="func">CharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FCA7A7A822BDA33"><span class="RefLink">71.3-1</span></a>) can be used for the cases 1. to 3.</p>
-
-<p><a id="X7FCA7A7A822BDA33" name="X7FCA7A7A822BDA33"></a></p>
-
-<h5>71.3-1 <span class="Heading">CharacterTable</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTable</code>( <var class="Arg">G</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTable</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTable</code>( <var class="Arg">name</var>[, <var class="Arg">param</var>] )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>Called with a group <var class="Arg">G</var>, <code class="func">CharacterTable</code> calls the attribute <code class="func">OrdinaryCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8011EEB684848039"><span class="RefLink">71.8-4</span></a>). Called with first argument a group <var class="Arg">G</var> or an ordinary character table <var class="Arg">ordtbl</var>, and second argument a prime <var class="Arg">p</var>, <code class="func">CharacterTable</code> calls the operation <code class="func">BrauerTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8476B25A79D7A7FC"><span class="RefLink">71.3-2</span></a>).</p>
-
-<p>Called with a string <var class="Arg">name</var> and perhaps optional parameters <var class="Arg">param</var>, <code class="func">CharacterTable</code> tries to access a character table from the <strong class="pkg">GAP</strong> Character Table Library. See the manual of the <strong class="pkg">GAP</strong> package <strong class="pkg">CTblLib</strong> for an overview of admissible arguments. An error is signalled if this <strong class="pkg">GAP</strong> package is not loaded in this case.</p>
-
-<p>Probably the most interesting information about the character table
-is its list of irreducibles, which can be accessed as the value of the
-attribute <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>). If the argument of <code class="func">CharacterTable</code> is a string <var class="Arg">name</var> then the irreducibles are just read from the library file, therefore the returned table stores them already. However, if <code class="func">CharacterTable</code> is called with a group <var class="Arg">G</var> or with an ordinary character table <var class="Arg">ordtbl</var>, the irreducible characters are <em>not</em> computed by <code class="func">CharacterTable</code>. They are only computed when the <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) value is accessed for the first time, for example when <code class="func">Display</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>) is called for the table (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C1941F17BE9FC21"><span class="RefLink">71.13</span></a>). This means for example that <code class="func">CharacterTable</code> returns its result very quickly, and the first call of <code class="func">Display</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>)
- for this table may take some time because the irreducible characters
-must be computed at that time before they can be displayed together with
- other information stored on the character table. The value of the
-filter <code class="code">HasIrr</code> indicates whether the irreducible characters have been computed already.</p>
-
-<p>The reason why <code class="func">CharacterTable</code> does not
-compute the irreducible characters is that there are situations where
-one only needs the "table head", that is, the information about class
-lengths, power maps etc., but not the irreducibles. For example, if one
-wants to inspect permutation characters of a group then all one has to
-do is to induce the trivial characters of subgroups one is interested
-in; for that, only class lengths and the class fusion are needed. Or if
-one wants to compute the Molien series (see <code class="func">MolienSeries</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X7D7F94D2820B1177"><span class="RefLink">72.12-1</span></a>)) for a given complex matrix group, the irreducible characters of this group are in general of no interest.</p>
-
-<p>For details about different algorithms to compute the irreducible characters, see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79BC08C6846718D9"><span class="RefLink">71.14</span></a>.</p>
-
-<p>If the group <var class="Arg">G</var> is given as an argument, <code class="func">CharacterTable</code> accesses the conjugacy classes of <var class="Arg">G</var> and therefore causes that these classes are computed if they were not yet stored (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X793E0EBF84B07313"><span class="RefLink">71.6</span></a>).</p>
-
-<p><a id="X8476B25A79D7A7FC" name="X8476B25A79D7A7FC"></a></p>
-
-<h5>71.3-2 <span class="Heading">BrauerTable</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ BrauerTable</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ BrauerTable</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ BrauerTableOp</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ComputedBrauerTables</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>Called with an ordinary character table <var class="Arg">ordtbl</var> or a group <var class="Arg">G</var>, <code class="func">BrauerTable</code> returns its <var class="Arg">p</var>-modular character table if <strong class="pkg">GAP</strong> can compute this table, and <code class="keyw">fail</code> otherwise. The <var class="Arg">p</var>-modular table can be computed for <var class="Arg">p</var>-solvable groups (using the Fong-Swan Theorem) and in the case that <var class="Arg">ordtbl</var> is a table from the <strong class="pkg">GAP</strong> character table library for which also the <var class="Arg">p</var>-modular table is contained in the table library.</p>
-
-<p>The default method for a group and a prime delegates to <code class="func">BrauerTable</code> for the ordinary character table of this group. The default method for <var class="Arg">ordtbl</var> uses the attribute <code class="func">ComputedBrauerTables</code> for storing the computed Brauer table at position <var class="Arg">p</var>, and calls the operation <code class="func">BrauerTableOp</code> for computing values that are not yet known.</p>
-
-<p>So if one wants to install a new method for computing Brauer tables then it is sufficient to install it for <code class="func">BrauerTableOp</code>.</p>
-
-<p>The <code class="keyw">mod</code> operator for a character table and a prime (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CADCBC9824CB624"><span class="RefLink">71.7</span></a>) delegates to <code class="func">BrauerTable</code>.</p>
-
-<p><a id="X85DB8AE7786A2DB5" name="X85DB8AE7786A2DB5"></a></p>
-
-<h5>71.3-3 CharacterTableRegular</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTableRegular</code>( <var class="Arg">tbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>For an ordinary character table <var class="Arg">tbl</var> and a prime integer <var class="Arg">p</var>, <code class="func">CharacterTableRegular</code> returns the "table head" of the <var class="Arg">p</var>-modular Brauer character table of <var class="Arg">tbl</var>. This is the restriction of <var class="Arg">tbl</var> to its <var class="Arg">p</var>-regular classes, like the return value of <code class="func">BrauerTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8476B25A79D7A7FC"><span class="RefLink">71.3-2</span></a>), but without the irreducible Brauer characters. (In general, these characters are hard to compute, and <code class="func">BrauerTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8476B25A79D7A7FC"><span class="RefLink">71.3-2</span></a>) may return <code class="keyw">fail</code> for the given arguments, for example if <var class="Arg">tbl</var> is a table from the <strong class="pkg">GAP</strong> character table library.)</p>
-
-<p>The returned table head can be used to create <var class="Arg">p</var>-modular
- Brauer characters, by restricting ordinary characters, for example when
- one is interested in approximations of the (unknown) irreducible Brauer
- characters.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">g:= SymmetricGroup( 4 );</span>
-Sym( [ 1 .. 4 ] )
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( g );; HasIrr( tbl );</span>
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">tblmod2:= CharacterTable( tbl, 2 );</span>
-BrauerTable( Sym( [ 1 .. 4 ] ), 2 )
-<span class="GAPprompt">gap></span> <span class="GAPinput">tblmod2 = CharacterTable( tbl, 2 );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">tblmod2 = BrauerTable( tbl, 2 );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">tblmod2 = BrauerTable( g, 2 );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">libtbl:= CharacterTable( "M" );</span>
-CharacterTable( "M" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTableRegular( libtbl, 2 );</span>
-BrauerTable( "M", 2 )
-<span class="GAPprompt">gap></span> <span class="GAPinput">BrauerTable( libtbl, 2 );</span>
-fail
-<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTable( "Symmetric", 4 );</span>
-CharacterTable( "Sym(4)" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">ComputedBrauerTables( tbl );</span>
-[ , BrauerTable( Sym( [ 1 .. 4 ] ), 2 ) ]
-</pre></div>
-
-<p><a id="X7DBEF4BF87F10CD6" name="X7DBEF4BF87F10CD6"></a></p>
-
-<h5>71.3-4 SupportedCharacterTableInfo</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SupportedCharacterTableInfo</code></td><td class="tdright">( global variable )</td></tr></tbody></table></div>
-<p><code class="func">SupportedCharacterTableInfo</code> is a list that contains at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>i</mi><mo>&#x2212;</mo><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-41-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.141em; display: inline-block;" role="math" id="MathJax-Span-165" class="math"><span style="display: inline-block; position: relative; width: 2.404em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.38em, 2.802em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-166" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-167" class="mn">3</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-168" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-169" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-170" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.081em; border-left: 0px solid; width: 0px; height: 1.022em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>i</mi><mo>−</mo><mn>2</mn></math></span></span><script id="MathJax-Element-41" type="math/tex">3i-2</script></span> an attribute getter function, at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>i</mi><mo>&#x2212;</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-42-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.141em; display: inline-block;" role="math" id="MathJax-Span-171" class="math"><span style="display: inline-block; position: relative; width: 2.404em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.3em, 2.802em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-172" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-173" class="mn">3</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-174" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-175" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-176" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.081em; border-left: 0px solid; width: 0px; height: 1.022em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>i</mi><mo>−</mo><mn>1</mn></math></span></span><script id="MathJax-Element-42" type="math/tex">3i-1</script></span> the name of this attribute, and at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-43-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-177" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.76em, 2.802em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-178" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-179" class="mn">3</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-180" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.081em; border-left: 0px solid; width: 0px; height: 1.022em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>i</mi></math></span></span><script id="MathJax-Element-43" type="math/tex">3i</script></span> a list containing one or two of the strings <code class="code">"class"</code>, <code class="code">"character"</code>,
- depending on whether the attribute value relies on the ordering of
-classes or characters. This allows one to set exactly the components
-with these names in the record that is later converted to the new table,
- in order to use the values as attribute values. So the record
-components that shall <em>not</em> be regarded as attribute values can be ignored. Also other attributes of the old table are ignored.</p>
-
-<p><code class="func">SupportedCharacterTableInfo</code> is used when (ordinary or Brauer) character table objects are created from records, using <code class="func">ConvertToCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8195BC057B1DFAD5"><span class="RefLink">71.3-5</span></a>).</p>
-
-<p>New attributes and properties can be notified to <code class="func">SupportedCharacterTableInfo</code> by creating them with <code class="code">DeclareAttributeSuppCT</code> and <code class="code">DeclarePropertySuppCT</code> instead of <code class="func">DeclareAttribute</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap79_mj.html#X7A00FC8A7A677A56"><span class="RefLink">79.18-3</span></a>) and <code class="func">DeclareProperty</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap79_mj.html#X7F4602F082682A04"><span class="RefLink">79.18-4</span></a>).</p>
-
-<p><a id="X8195BC057B1DFAD5" name="X8195BC057B1DFAD5"></a></p>
-
-<h5>71.3-5 ConvertToCharacterTable</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ConvertToCharacterTable</code>( <var class="Arg">record</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ConvertToCharacterTableNC</code>( <var class="Arg">record</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">record</var> be a record. <code class="func">ConvertToCharacterTable</code> converts <var class="Arg">record</var> into a component object (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap79_mj.html#X866E223484649E5A"><span class="RefLink">79.10</span></a>) representing a character table. The values of those components of <var class="Arg">record</var> whose names occur in <code class="func">SupportedCharacterTableInfo</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7DBEF4BF87F10CD6"><span class="RefLink">71.3-4</span></a>)
- correspond to attribute values of the returned character table. All
-other components of the record simply become components of the character
- table object.</p>
-
-<p>If inconsistencies in <var class="Arg">record</var> are detected, <code class="keyw">fail</code> is returned. <var class="Arg">record</var> must have the component <code class="code">UnderlyingCharacteristic</code> bound (cf. <code class="func">UnderlyingCharacteristic</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7F58A82F7D88000A"><span class="RefLink">71.9-5</span></a>)), since this decides about whether the returned character table lies in <code class="func">IsOrdinaryTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X82FF82C87CF82ADF"><span class="RefLink">71.4-1</span></a>) or in <code class="func">IsBrauerTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X82FF82C87CF82ADF"><span class="RefLink">71.4-1</span></a>).</p>
-
-<p><code class="func">ConvertToCharacterTableNC</code> does the same except that all checks of <var class="Arg">record</var> are omitted.</p>
-
-<p>An example of a conversion from a record to a character table object can be found in Section <code class="func">PrintCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79EC9603833AA2AB"><span class="RefLink">71.13-5</span></a>).</p>
-
-<p><a id="X789FAC077AEF088A" name="X789FAC077AEF088A"></a></p>
-
-<h4>71.4 <span class="Heading">Character Table Categories</span></h4>
-
-<p><a id="X82FF82C87CF82ADF" name="X82FF82C87CF82ADF"></a></p>
-
-<h5>71.4-1 IsNearlyCharacterTable</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsNearlyCharacterTable</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsCharacterTable</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsOrdinaryTable</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsBrauerTable</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsCharacterTableInProgress</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></tbody></table></div>
-<p>Every "character table like object" in <strong class="pkg">GAP</strong> lies in the category <code class="func">IsNearlyCharacterTable</code>. There are four important subcategories, namely the <em>ordinary</em> tables in <code class="func">IsOrdinaryTable</code>, the <em>Brauer</em> tables in <code class="func">IsBrauerTable</code>, the union of these two in <code class="func">IsCharacterTable</code>, and the <em>incomplete ordinary</em> tables in <code class="func">IsCharacterTableInProgress</code>.</p>
-
-<p>We want to distinguish ordinary and Brauer tables because a Brauer
-table may delegate tasks to the ordinary table of the same group, for
-example the computation of power maps. A Brauer table is constructed
-from an ordinary table and stores this table upon construction
-(see <code class="func">OrdinaryCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8011EEB684848039"><span class="RefLink">71.8-4</span></a>)).</p>
-
-<p>Furthermore, <code class="func">IsOrdinaryTable</code> and <code class="func">IsBrauerTable</code>
- denote character tables that provide enough information to compute all
-power maps and irreducible characters (and in the case of Brauer tables
-to get the ordinary table), for example because the underlying group
-(see <code class="func">UnderlyingGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FF4826A82B667AF"><span class="RefLink">71.6-1</span></a>)) is known or because the table is a library table (see the manual of the <strong class="pkg">GAP</strong>
- Character Table Library). We want to distinguish these tables from
-partially known ordinary tables that cannot be asked for all power maps
-or all irreducible characters.</p>
-
-<p>The character table objects in <code class="func">IsCharacterTable</code> are always immutable (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap12_mj.html#X7F0C119682196D65"><span class="RefLink">12.6</span></a>).
- This means mainly that the ordering of conjugacy classes used for the
-various attributes of the character table cannot be changed; see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X816FCD5A805F9FE8"><span class="RefLink">71.21</span></a> for how to compute a character table with a different ordering of classes.</p>
-
-<p>The <strong class="pkg">GAP</strong> objects in <code class="func">IsCharacterTableInProgress</code>
- represent incomplete ordinary character tables. This means that not all
- irreducible characters, not all power maps are known, and perhaps even
-the number of classes and the centralizer orders are known. Such tables
-occur when the character table of a group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-44-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-181" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-182" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-183" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-44" type="math/tex">G</script></span> is constructed using character tables of related groups and information about <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-45-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-184" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-185" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-186" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-45" type="math/tex">G</script></span> but for example without explicitly computing the conjugacy classes of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-46-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-187" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-188" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-189" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-46" type="math/tex">G</script></span>. An object in <code class="func">IsCharacterTableInProgress</code> is first of all <em>mutable</em>, so <em>nothing is stored automatically</em>
- on such a table, since otherwise one has no control of side-effects
-when a hypothesis is changed. Operations for such tables may return more
- general values than for other tables, for example class functions may
-contain unknowns (see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap74_mj.html#X7C1FAB6280A02CCB"><span class="RefLink">74</span></a>) or lists of possible values in certain positions, the same may happen also for power maps and class fusions (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X7F18772E86F06179"><span class="RefLink">73.5</span></a>). <em>Incomplete
- tables in this sense are currently not supported and will be described
-in a chapter of their own when they become available.</em> Note that the term "incomplete table" shall express that <strong class="pkg">GAP</strong>
- cannot compute certain values such as irreducible characters or power
-maps. A table with access to its group is therefore always complete,
-also if its irreducible characters are not yet stored.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">g:= SymmetricGroup( 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( g ); modtbl:= tbl mod 2;</span>
-CharacterTable( Sym( [ 1 .. 4 ] ) )
-BrauerTable( Sym( [ 1 .. 4 ] ), 2 )
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsCharacterTable( tbl ); IsCharacterTable( modtbl );</span>
-true
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsBrauerTable( modtbl ); IsBrauerTable( tbl );</span>
-true
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsOrdinaryTable( tbl ); IsOrdinaryTable( modtbl );</span>
-true
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsCharacterTable( g ); IsCharacterTable( Irr( g ) );</span>
-false
-false
-</pre></div>
-
-<p><a id="X7C6F3D947E5188D1" name="X7C6F3D947E5188D1"></a></p>
-
-<h5>71.4-2 InfoCharacterTable</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ InfoCharacterTable</code></td><td class="tdright">( info class )</td></tr></tbody></table></div>
-<p>is the info class (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap7_mj.html#X7A9C902479CB6F7C"><span class="RefLink">7.4</span></a>) for computations with character tables.</p>
-
-<p><a id="X7FA867637EBB30F9" name="X7FA867637EBB30F9"></a></p>
-
-<h5>71.4-3 NearlyCharacterTablesFamily</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ NearlyCharacterTablesFamily</code></td><td class="tdright">( family )</td></tr></tbody></table></div>
-<p>Every character table like object lies in this family (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap13_mj.html#X846063757EC05986"><span class="RefLink">13.1</span></a>).</p>
-
-<p><a id="X829C4B6E83998F40" name="X829C4B6E83998F40"></a></p>
-
-<h4>71.5 <span class="Heading">Conventions for Character Tables</span></h4>
-
-<p>The following few conventions should be noted.</p>
-
-
-<ul>
-<li><p>The class of the <em>identity element</em> is expected to be the first one; thus the degree of a character is the character value at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-47-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-190" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.38em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-191" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-192" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script id="MathJax-Element-47" type="math/tex">1</script></span>.</p>
-
-</li>
-<li><p>The <em>trivial character</em> of a character table need not be the first in the list of irreducibles.</p>
-
-</li>
-<li><p>Most functions that take a character table as an argument and
-work with characters expect these characters as an argument, too. For
-some functions, the list of irreducible characters serves as the
-default, i.e, the value of the attribute <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>); in these cases, the <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) value is automatically computed if it was not yet known.</p>
-
-</li>
-<li><p>For a stored class fusion, the image table is denoted by its <code class="func">Identifier</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79C40EE97890202F"><span class="RefLink">71.9-8</span></a>) value; each library table has a unique identifier by which it can be accessed (see <a href="http://www.gap-system.org/Manuals/pkg/ctbllib/doc/chap2.html#X84930B2D7849E019"><span class="RefLink">CTblLib: Accessing a Character Table from the Library</span></a> in the manual for the <strong class="pkg">GAP</strong> Character Table Library), tables constructed from groups get an identifier that is unique in the current <strong class="pkg">GAP</strong> session.</p>
-
-</li>
-</ul>
-<p><a id="X793E0EBF84B07313" name="X793E0EBF84B07313"></a></p>
-
-<h4>71.6 <span class="Heading">The Interface between Character Tables and Groups</span></h4>
-
-<p>For a character table with underlying group (see <code class="func">UnderlyingGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FF4826A82B667AF"><span class="RefLink">71.6-1</span></a>)), the interface between table and group consists of three attribute values, namely the <em>group</em>, the <em>conjugacy classes</em> stored in the table (see <code class="func">ConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X849A38F887F6EC86"><span class="RefLink">71.6-2</span></a>) below) and the <em>identification</em> of the conjugacy classes of table and group (see <code class="func">IdentificationOfConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X84DC12AA804C8085"><span class="RefLink">71.6-3</span></a>) below).</p>
-
-<p>Character tables constructed from groups know these values upon
-construction, and for character tables constructed without groups, these
- values are usually not known and cannot be computed from the table.</p>
-
-<p>However, given a group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-48-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-193" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-194" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-195" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-48" type="math/tex">G</script></span> and a character table of a group isomorphic to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-49-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-196" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-197" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-198" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-49" type="math/tex">G</script></span> (for example a character table from the <strong class="pkg">GAP</strong> table library), one can tell <strong class="pkg">GAP</strong> to compute a new instance of the given table and to use it as the character table of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-50-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-199" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-200" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-201" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-50" type="math/tex">G</script></span> (see <code class="func">CharacterTableWithStoredGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8788C6C7829C1ADE"><span class="RefLink">71.6-4</span></a>)).</p>
-
-<p>Tasks may be delegated from a group to its character table or vice
-versa only if these three attribute values are stored in the character
-table.</p>
-
-<p><a id="X7FF4826A82B667AF" name="X7FF4826A82B667AF"></a></p>
-
-<h5>71.6-1 UnderlyingGroup</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ UnderlyingGroup</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For an ordinary character table <var class="Arg">ordtbl</var> of a finite group, the group can be stored as value of <code class="func">UnderlyingGroup</code>.</p>
-
-<p>Brauer tables do not store the underlying group, they access it via the ordinary table (see <code class="func">OrdinaryCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8011EEB684848039"><span class="RefLink">71.8-4</span></a>)).</p>
-
-<p><a id="X849A38F887F6EC86" name="X849A38F887F6EC86"></a></p>
-
-<h5>71.6-2 ConjugacyClasses</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ConjugacyClasses</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For a character table <var class="Arg">tbl</var> with known underlying group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-51-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-202" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-203" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-204" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-51" type="math/tex">G</script></span>, the <code class="func">ConjugacyClasses</code> value of <var class="Arg">tbl</var> is a list of conjugacy classes of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-52-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-205" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-206" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-207" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-52" type="math/tex">G</script></span>.
- All those lists stored in the table that are related to the ordering of
- conjugacy classes (such as sizes of centralizers and conjugacy classes,
- orders of representatives, power maps, and all class functions) refer
-to the ordering of this list.</p>
-
-<p>This ordering need <em>not</em> coincide with the ordering of conjugacy classes as stored in the underlying group of the table (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X816FCD5A805F9FE8"><span class="RefLink">71.21</span></a>).
- One reason for this is that otherwise we would not be allowed to use a
-library table as the character table of a group for which the conjugacy
-classes are stored already. (Another, less important reason is that we
-can use the same group as underlying group of character tables that
-differ only w.r.t. the ordering of classes.)</p>
-
-<p>The class of the identity element must be the first class (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X829C4B6E83998F40"><span class="RefLink">71.5</span></a>).</p>
-
-<p>If <var class="Arg">tbl</var> was constructed from <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-53-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-208" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-209" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-210" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-53" type="math/tex">G</script></span> then the conjugacy classes have been stored at the same time when <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-54-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-211" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-212" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-213" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-54" type="math/tex">G</script></span> was stored. If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-55-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-214" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-215" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-216" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-55" type="math/tex">G</script></span> and <var class="Arg">tbl</var> have been connected later than in the construction of <var class="Arg">tbl</var>, the recommended way to do this is via <code class="func">CharacterTableWithStoredGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8788C6C7829C1ADE"><span class="RefLink">71.6-4</span></a>). So there is no method for <code class="func">ConjugacyClasses</code> that computes the value for <var class="Arg">tbl</var> if it is not yet stored.</p>
-
-<p>Brauer tables do not store the (<span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-56-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-217" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-218" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-219" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-56" type="math/tex">p</script></span>-regular) conjugacy classes, they access them via the ordinary table (see <code class="func">OrdinaryCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8011EEB684848039"><span class="RefLink">71.8-4</span></a>)) if necessary.</p>
-
-<p><a id="X84DC12AA804C8085" name="X84DC12AA804C8085"></a></p>
-
-<h5>71.6-3 IdentificationOfConjugacyClasses</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IdentificationOfConjugacyClasses</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For an ordinary character table <var class="Arg">tbl</var> with known underlying group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-57-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-220" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-221" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-222" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-57" type="math/tex">G</script></span>, <code class="func">IdentificationOfConjugacyClasses</code> returns a list of positive integers that contains at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-58-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-223" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-224" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-225" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-58" type="math/tex">i</script></span> the position of the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-59-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-226" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-227" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-228" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-59" type="math/tex">i</script></span>-th conjugacy class of <var class="Arg">tbl</var> in the <code class="func">ConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X849A38F887F6EC86"><span class="RefLink">71.6-2</span></a>) value of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-60-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-229" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-230" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-231" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-60" type="math/tex">G</script></span>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">g:= SymmetricGroup( 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">repres:= [ (1,2), (1,2,3), (1,2,3,4), (1,2)(3,4), () ];;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ccl:= List( repres, x -> ConjugacyClass( g, x ) );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">SetConjugacyClasses( g, ccl );</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( g );; # the table stores already the values</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">HasConjugacyClasses( tbl ); HasUnderlyingGroup( tbl );</span>
-true
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingGroup( tbl ) = g;</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">HasIdentificationOfConjugacyClasses( tbl );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">IdentificationOfConjugacyClasses( tbl );</span>
-[ 5, 1, 2, 3, 4 ]
-</pre></div>
-
-<p><a id="X8788C6C7829C1ADE" name="X8788C6C7829C1ADE"></a></p>
-
-<h5>71.6-4 CharacterTableWithStoredGroup</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTableWithStoredGroup</code>( <var class="Arg">G</var>, <var class="Arg">tbl</var>[, <var class="Arg">info</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">G</var> be a group and <var class="Arg">tbl</var> a character table of (a group isomorphic to) <var class="Arg">G</var>, such that <var class="Arg">G</var> does not store its <code class="func">OrdinaryCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8011EEB684848039"><span class="RefLink">71.8-4</span></a>) value. <code class="func">CharacterTableWithStoredGroup</code> calls <code class="func">CompatibleConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X790019E87CFDDB98"><span class="RefLink">71.6-5</span></a>), trying to identify the classes of <var class="Arg">G</var> with the columns of <var class="Arg">tbl</var>.</p>
-
-<p>If this identification is unique up to automorphisms of <var class="Arg">tbl</var> (see <code class="func">AutomorphismsOfTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C2753DE8094F4BA"><span class="RefLink">71.9-4</span></a>)) then <var class="Arg">tbl</var> is stored as <code class="func">CharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FCA7A7A822BDA33"><span class="RefLink">71.3-1</span></a>) value of <var class="Arg">G</var>, and a new character table is returned that is equivalent to <var class="Arg">tbl</var>, is sorted in the same way as <var class="Arg">tbl</var>, and has the values of <code class="func">UnderlyingGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FF4826A82B667AF"><span class="RefLink">71.6-1</span></a>), <code class="func">ConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X849A38F887F6EC86"><span class="RefLink">71.6-2</span></a>), and <code class="func">IdentificationOfConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X84DC12AA804C8085"><span class="RefLink">71.6-3</span></a>) set.</p>
-
-<p>Otherwise, i.e., if <strong class="pkg">GAP</strong> cannot identify the classes of <var class="Arg">G</var> up to automorphisms of <var class="Arg">tbl</var>, <code class="keyw">fail</code> is returned.</p>
-
-<p>If a record is present as the third argument <var class="Arg">info</var>, its meaning is the same as the optional argument <var class="Arg">arec</var> for <code class="func">CompatibleConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X790019E87CFDDB98"><span class="RefLink">71.6-5</span></a>).</p>
-
-<p>If a list is entered as third argument <var class="Arg">info</var> it is used as value of <code class="func">IdentificationOfConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X84DC12AA804C8085"><span class="RefLink">71.6-3</span></a>), relative to the <code class="func">ConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X849A38F887F6EC86"><span class="RefLink">71.6-2</span></a>) value of <var class="Arg">G</var>, without further checking, and the corresponding character table is returned.</p>
-
-<p><a id="X790019E87CFDDB98" name="X790019E87CFDDB98"></a></p>
-
-<h5>71.6-5 CompatibleConjugacyClasses</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CompatibleConjugacyClasses</code>( [<var class="Arg">G</var>, <var class="Arg">ccl</var>, ]<var class="Arg">tbl</var>[, <var class="Arg">arec</var>] )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>If the arguments <var class="Arg">G</var> and <var class="Arg">ccl</var> are present then <var class="Arg">ccl</var> must be a list of the conjugacy classes of the group <var class="Arg">G</var>, and <var class="Arg">tbl</var> the ordinary character table of <var class="Arg">G</var>. Then <code class="func">CompatibleConjugacyClasses</code> returns a list <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-61-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-232" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.29em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-233" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-234" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math></span></span><script id="MathJax-Element-61" type="math/tex">l</script></span> of positive integers that describes an identification of the columns of <var class="Arg">tbl</var> with the conjugacy classes <var class="Arg">ccl</var> in the sense that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-62-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.641em; display: inline-block;" role="math" id="MathJax-Span-235" class="math"><span style="display: inline-block; position: relative; width: 1.25em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1001.16em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-236" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-237" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-238" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-239" class="mi">i</span><span style="font-family: STIXGeneral;" id="MathJax-Span-240" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.216em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-62" type="math/tex">l[i]</script></span> is the position in <var class="Arg">ccl</var> of the class corresponding to the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-63-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-241" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-242" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-243" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-63" type="math/tex">i</script></span>-th column of <var class="Arg">tbl</var>, if this identification is unique up to automorphisms of <var class="Arg">tbl</var> (see <code class="func">AutomorphismsOfTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C2753DE8094F4BA"><span class="RefLink">71.9-4</span></a>)); if <strong class="pkg">GAP</strong> cannot identify the classes, <code class="keyw">fail</code> is returned.</p>
-
-<p>If <var class="Arg">tbl</var> is the first argument then it must be an ordinary character table, and <code class="func">CompatibleConjugacyClasses</code> checks whether the columns of <var class="Arg">tbl</var> can be identified with the conjugacy classes of a group isomorphic to that for which <var class="Arg">tbl</var> is the character table; the return value is a list of all those sets of class positions for which the columns of <var class="Arg">tbl</var> cannot be distinguished with the invariants used, up to automorphisms of <var class="Arg">tbl</var>. So the identification is unique if and only if the returned list is empty.</p>
-
-<p>The usual approach is that one first calls <code class="func">CompatibleConjugacyClasses</code>
- in the second form for checking quickly whether the first form will be
-successful, and only if this is the case the more time consuming
-calculations with both group and character table are done.</p>
-
-<p>The following invariants are used.</p>
-
-<ol>
-<li><p>element orders (see <code class="func">OrdersClassRepresentatives</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>)),</p>
-
-</li>
-<li><p>class lengths (see <code class="func">SizesConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D9D2A45879A6A62"><span class="RefLink">71.9-3</span></a>)),</p>
-
-</li>
-<li><p>power maps (see <code class="func">PowerMap</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>), <code class="func">ComputedPowerMaps</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>)),</p>
-
-</li>
-<li><p>symmetries of the table (see <code class="func">AutomorphismsOfTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C2753DE8094F4BA"><span class="RefLink">71.9-4</span></a>)).</p>
-
-</li>
-</ol>
-<p>If the optional argument <var class="Arg">arec</var> is present then
-it must be a record whose components describe additional information for
- the class identification. The following components are supported.</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code">natchar</code> </strong></dt>
-<dd><p>if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-64-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-244" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-245" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-246" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-64" type="math/tex">G</script></span>
- is a permutation group or matrix group then the value of this component
- is regarded as the list of values of the natural character (see <code class="func">NaturalCharacter</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X82C01DDB82D751A9"><span class="RefLink">72.7-2</span></a>)) of <var class="Arg">G</var>, w.r.t. the ordering of classes in <var class="Arg">tbl</var>,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">bijection</code> </strong></dt>
-<dd><p>a list describing a partial bijection; the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-65-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-247" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-248" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-249" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-65" type="math/tex">i</script></span>-th entry, if bound, is the position of the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-66-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-250" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-251" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-252" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-66" type="math/tex">i</script></span>-th conjugacy class of <var class="Arg">tbl</var> in the list <var class="Arg">ccl</var>.</p>
-
-</dd>
-</dl>
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">g:= AlternatingGroup( 5 );</span>
-Alt( [ 1 .. 5 ] )
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );</span>
-CharacterTable( "A5" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">HasUnderlyingGroup( tbl ); HasOrdinaryCharacterTable( g );</span>
-false
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">CompatibleConjugacyClasses( tbl ); # unique identification</span>
-[ ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">new:= CharacterTableWithStoredGroup( g, tbl );</span>
-CharacterTable( Alt( [ 1 .. 5 ] ) )
-<span class="GAPprompt">gap></span> <span class="GAPinput">Irr( new ) = Irr( tbl );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">HasConjugacyClasses( new ); HasUnderlyingGroup( new );</span>
-true
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">IdentificationOfConjugacyClasses( new );</span>
-[ 1, 2, 3, 4, 5 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput"># Here is an example where the identification is not unique.</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">CompatibleConjugacyClasses( CharacterTable( "J2" ) );</span>
-[ [ 17, 18 ], [ 9, 10 ] ]
-</pre></div>
-
-<p><a id="X7CADCBC9824CB624" name="X7CADCBC9824CB624"></a></p>
-
-<h4>71.7 <span class="Heading">Operators for Character Tables</span></h4>
-
-<p>The following infix operators are defined for character tables.</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code"><var class="Arg">tbl1</var> * <var class="Arg">tbl2</var></code></strong></dt>
-<dd><p>the direct product of two character tables (see <code class="func">CharacterTableDirectProduct</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7BE1572D7BBC6AC8"><span class="RefLink">71.20-1</span></a>)),</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code"><var class="Arg">tbl</var> / <var class="Arg">list</var></code></strong></dt>
-<dd><p>the table of the factor group modulo the normal subgroup spanned by the classes in the list <var class="Arg">list</var> (see <code class="func">CharacterTableFactorGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C3A4E5283B240BE"><span class="RefLink">71.20-3</span></a>)),</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code"><var class="Arg">tbl</var> mod <var class="Arg">p</var></code></strong></dt>
-<dd><p>the <var class="Arg">p</var>-modular Brauer character table corresponding to the ordinary character table <var class="Arg">tbl</var> (see <code class="func">BrauerTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8476B25A79D7A7FC"><span class="RefLink">71.3-2</span></a>)),</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code"><var class="Arg">tbl</var>.<var class="Arg">name</var></code></strong></dt>
-<dd><p>the position of the class with name <var class="Arg">name</var> in <var class="Arg">tbl</var> (see <code class="func">ClassNames</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X804CFD597C795801"><span class="RefLink">71.9-6</span></a>)).</p>
-
-</dd>
-</dl>
-<p><a id="X7F9D58208241D35E" name="X7F9D58208241D35E"></a></p>
-
-<h4>71.8 <span class="Heading">Attributes and Properties for Groups and Character Tables</span></h4>
-
-<p>Several <em>attributes for groups</em> are valid also for character tables.</p>
-
-<p>These are first those that have the same meaning for both the group
-and its character table, and whose values can be read off or computed,
-respectively, from the character table, such as <code class="func">Size</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X81EFD9FE804AC6EE"><span class="RefLink">71.8-5</span></a>), <code class="func">IsAbelian</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X81EFD9FE804AC6EE"><span class="RefLink">71.8-5</span></a>), or <code class="func">IsSolvable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X81EFD9FE804AC6EE"><span class="RefLink">71.8-5</span></a>).</p>
-
-<p>Second, there are attributes whose meaning for character tables is different from the meaning for groups, such as <code class="func">ConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X849A38F887F6EC86"><span class="RefLink">71.6-2</span></a>).</p>
-
-<p><a id="X81FEFF768134481A" name="X81FEFF768134481A"></a></p>
-
-<h5>71.8-1 <span class="Heading">CharacterDegrees</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterDegrees</code>( <var class="Arg">G</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterDegrees</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>In the first form, <code class="func">CharacterDegrees</code> returns a collected list of the degrees of the absolutely irreducible characters of the group <var class="Arg">G</var>; the optional second argument <var class="Arg">p</var> must be either zero or a prime integer denoting the characteristic, the default value is zero. In the second form, <var class="Arg">tbl</var> must be an (ordinary or Brauer) character table, and <code class="func">CharacterDegrees</code> returns a collected list of the degrees of the absolutely irreducible characters of <var class="Arg">tbl</var>.</p>
-
-<p>(The default method for the call with only argument a group is to call the operation with second argument <code class="code">0</code>.)</p>
-
-<p>For solvable groups, the default method is based on <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCon90b">[Con90b]</a>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">CharacterDegrees( SymmetricGroup( 4 ) );</span>
-[ [ 1, 2 ], [ 2, 1 ], [ 3, 2 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterDegrees( SymmetricGroup( 4 ), 2 );</span>
-[ [ 1, 1 ], [ 2, 1 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterDegrees( CharacterTable( "A5" ) );</span>
-[ [ 1, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5, 1 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterDegrees( CharacterTable( "A5" ) mod 2 );</span>
-[ [ 1, 1 ], [ 2, 2 ], [ 4, 1 ] ]
-</pre></div>
-
-<p><a id="X873B3CC57E9A5492" name="X873B3CC57E9A5492"></a></p>
-
-<h5>71.8-2 <span class="Heading">Irr</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Irr</code>( <var class="Arg">G</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Irr</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>Called with a group <var class="Arg">G</var>, <code class="func">Irr</code> returns the irreducible characters of the ordinary character table of <var class="Arg">G</var>. Called with a group <var class="Arg">G</var> and a prime integer <var class="Arg">p</var>, <code class="func">Irr</code> returns the irreducible characters of the <var class="Arg">p</var>-modular Brauer table of <var class="Arg">G</var>. Called with an (ordinary or Brauer) character table <var class="Arg">tbl</var>, <code class="func">Irr</code> returns the list of all complex absolutely irreducible characters of <var class="Arg">tbl</var>.</p>
-
-<p>For a character table <var class="Arg">tbl</var> with underlying group, <code class="func">Irr</code> may delegate to the group. For a group <var class="Arg">G</var>, <code class="func">Irr</code> may delegate to its character table only if the irreducibles are already stored there.</p>
-
-<p>(If <var class="Arg">G</var> is <var class="Arg">p</var>-solvable (see <code class="func">IsPSolvable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X81130F9A7CFCF6BF"><span class="RefLink">39.15-23</span></a>)) then the <var class="Arg">p</var>-modular irreducible characters can be computed by the Fong-Swan Theorem; in all other cases, there may be no method.)</p>
-
-<p>Note that the ordering of columns in the <code class="func">Irr</code> matrix of the group <var class="Arg">G</var> refers to the ordering of conjugacy classes in the <code class="func">CharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FCA7A7A822BDA33"><span class="RefLink">71.3-1</span></a>) value of <var class="Arg">G</var>, which may differ from the ordering of conjugacy classes in <var class="Arg">G</var> (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X793E0EBF84B07313"><span class="RefLink">71.6</span></a>). As an extreme example, for a character table obtained from sorting the classes of the <code class="func">CharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FCA7A7A822BDA33"><span class="RefLink">71.3-1</span></a>) value of <var class="Arg">G</var>, the ordering of columns in the <code class="func">Irr</code> matrix respects the sorting of classes (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X816FCD5A805F9FE8"><span class="RefLink">71.21</span></a>), so the irreducibles of such a table will in general not coincide with the irreducibles stored as the <code class="func">Irr</code> value of <var class="Arg">G</var> although also the sorted table stores the group <var class="Arg">G</var>.</p>
-
-<p>The ordering of the entries in the attribute <code class="func">Irr</code> of a group need <em>not</em> coincide with the ordering of its <code class="func">IrreducibleRepresentations</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7F29C5447B5DC102"><span class="RefLink">71.14-4</span></a>) value.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">Irr( SymmetricGroup( 4 ) );</span>
-[ Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, -1, 1, 1, -1
- ] ), Character( CharacterTable( Sym( [ 1 .. 4 ] ) ),
- [ 3, -1, -1, 0, 1 ] ),
- Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 2, 0, 2, -1, 0 ] )
- , Character( CharacterTable( Sym( [ 1 .. 4 ] ) ),
- [ 3, 1, -1, 0, -1 ] ),
- Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1, 1 ] )
- ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Irr( SymmetricGroup( 4 ), 2 );</span>
-[ Character( BrauerTable( Sym( [ 1 .. 4 ] ), 2 ), [ 1, 1 ] ),
- Character( BrauerTable( Sym( [ 1 .. 4 ] ), 2 ), [ 2, -1 ] ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Irr( CharacterTable( "A5" ) );</span>
-[ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ),
- Character( CharacterTable( "A5" ),
- [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ),
- Character( CharacterTable( "A5" ),
- [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ),
- Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ),
- Character( CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Irr( CharacterTable( "A5" ) mod 2 );</span>
-[ Character( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] ),
- Character( BrauerTable( "A5", 2 ),
- [ 2, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ] ),
- Character( BrauerTable( "A5", 2 ),
- [ 2, -1, E(5)^2+E(5)^3, E(5)+E(5)^4 ] ),
- Character( BrauerTable( "A5", 2 ), [ 4, 1, -1, -1 ] ) ]
-</pre></div>
-
-<p><a id="X8549899A7DE206BA" name="X8549899A7DE206BA"></a></p>
-
-<h5>71.8-3 <span class="Heading">LinearCharacters</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ LinearCharacters</code>( <var class="Arg">G</var>[, <var class="Arg">p</var>] )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ LinearCharacters</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p><code class="func">LinearCharacters</code> returns the linear (i.e., degree <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-67-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-253" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.38em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-254" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-255" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script id="MathJax-Element-67" type="math/tex">1</script></span>) characters in the <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) list of the group <var class="Arg">G</var> or the character table <var class="Arg">tbl</var>, respectively. In the second form, <code class="func">LinearCharacters</code> returns the <var class="Arg">p</var>-modular linear characters of the group <var class="Arg">G</var>.</p>
-
-<p>For a character table <var class="Arg">tbl</var> with underlying group, <code class="func">LinearCharacters</code> may delegate to the group. For a group <var class="Arg">G</var>, <code class="func">LinearCharacters</code> may delegate to its character table only if the irreducibles are already stored there.</p>
-
-<p>The ordering of linear characters in <var class="Arg">tbl</var> need not coincide with the ordering of linear characters in the irreducibles of <var class="Arg">tbl</var> (see <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>)).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">LinearCharacters( SymmetricGroup( 4 ) );</span>
-[ Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1, 1 ] ),
- Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, -1, 1, 1, -1
- ] ) ]
-</pre></div>
-
-<p><a id="X8011EEB684848039" name="X8011EEB684848039"></a></p>
-
-<h5>71.8-4 <span class="Heading">OrdinaryCharacterTable</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ OrdinaryCharacterTable</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ OrdinaryCharacterTable</code>( <var class="Arg">modtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p><code class="func">OrdinaryCharacterTable</code> returns the ordinary character table of the group <var class="Arg">G</var> or the Brauer character table <var class="Arg">modtbl</var>, respectively.</p>
-
-<p>Since Brauer character tables are constructed from ordinary tables, the attribute value for <var class="Arg">modtbl</var> is already stored (cf. <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X789FAC077AEF088A"><span class="RefLink">71.4</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">OrdinaryCharacterTable( SymmetricGroup( 4 ) );</span>
-CharacterTable( Sym( [ 1 .. 4 ] ) )
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );; modtbl:= tbl mod 2;</span>
-BrauerTable( "A5", 2 )
-<span class="GAPprompt">gap></span> <span class="GAPinput">OrdinaryCharacterTable( modtbl ) = tbl;</span>
-true
-</pre></div>
-
-<p><a id="X81EFD9FE804AC6EE" name="X81EFD9FE804AC6EE"></a></p>
-
-<h5>71.8-5 <span class="Heading">Group Operations Applicable to Character Tables</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ AbelianInvariants</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CommutatorLength</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Exponent</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsAbelian</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsAlmostSimple</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsCyclic</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsElementaryAbelian</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsFinite</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsMonomial</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsNilpotent</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsPerfect</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsSimple</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsSolvable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsSporadicSimple</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsSupersolvable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsomorphismTypeInfoFiniteSimpleGroup</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ NrConjugacyClasses</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Size</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>These operations for groups are applicable to character tables and
-mean the same for a character table as for its underlying group; see
-Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X8716635F7951801B"><span class="RefLink">39</span></a>
- for the definitions. The operations are mainly useful for selecting
-character tables with certain properties, also for character tables
-without access to a group.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tables:= [ CharacterTable( CyclicGroup( 3 ) ),</span>
-<span class="GAPprompt">></span> <span class="GAPinput"> CharacterTable( SymmetricGroup( 4 ) ),</span>
-<span class="GAPprompt">></span> <span class="GAPinput"> CharacterTable( AlternatingGroup( 5 ) ) ];;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, AbelianInvariants );</span>
-[ [ 3 ], [ 2 ], [ ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, CommutatorLength );</span>
-[ 1, 1, 1 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, Exponent );</span>
-[ 3, 12, 30 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsAbelian );</span>
-[ true, false, false ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsAlmostSimple );</span>
-[ false, false, true ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsCyclic );</span>
-[ true, false, false ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsFinite );</span>
-[ true, true, true ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsMonomial );</span>
-[ true, true, false ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsNilpotent );</span>
-[ true, false, false ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsPerfect );</span>
-[ false, false, true ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsSimple );</span>
-[ true, false, true ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsSolvable );</span>
-[ true, true, false ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsSupersolvable );</span>
-[ true, false, false ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, NrConjugacyClasses );</span>
-[ 3, 5, 5 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, Size );</span>
-[ 3, 24, 60 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( CharacterTable( "C5" ) );</span>
-rec( name := "Z(5)", parameter := 5, series := "Z" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( CharacterTable( "S3" ) );</span>
-fail
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( CharacterTable( "S6(3)" ) );</span>
-rec( name := "C(3,3) = S(6,3)", parameter := [ 3, 3 ], series := "C" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( CharacterTable( "O7(3)" ) );</span>
-rec( name := "B(3,3) = O(7,3)", parameter := [ 3, 3 ], series := "B" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( CharacterTable( "A8" ) );</span>
-rec( name := "A(8) ~ A(3,2) = L(4,2) ~ D(3,2) = O+(6,2)",
- parameter := 8, series := "A" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( CharacterTable( "L3(4)" ) );</span>
-rec( name := "A(2,4) = L(3,4)", parameter := [ 3, 4 ], series := "L" )
-</pre></div>
-
-<p><a id="X7995A2AD83BC58A0" name="X7995A2AD83BC58A0"></a></p>
-
-<h4>71.9 <span class="Heading">Attributes and Properties only for Character Tables</span></h4>
-
-<p>The following three <em>attributes for character tables</em> –<code class="func">OrdersClassRepresentatives</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>), <code class="func">SizesCentralizers</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CF7907F790A5DE6"><span class="RefLink">71.9-2</span></a>), and <code class="func">SizesConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D9D2A45879A6A62"><span class="RefLink">71.9-3</span></a>)– would make sense also for groups but are in fact <em>not</em> used for groups. This is because the values depend on the ordering of conjugacy classes stored as the value of <code class="func">ConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X849A38F887F6EC86"><span class="RefLink">71.6-2</span></a>), and this value may differ for a group and its character table (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X793E0EBF84B07313"><span class="RefLink">71.6</span></a>).
- Note that for character tables, the consistency of attribute values
-must be guaranteed, whereas for groups, there is no need to impose such a
- consistency rule.</p>
-
-<p>The other attributes introduced in this section apply only to character tables, not to groups.</p>
-
-<p><a id="X86F455DA7A9C30EE" name="X86F455DA7A9C30EE"></a></p>
-
-<h5>71.9-1 OrdersClassRepresentatives</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ OrdersClassRepresentatives</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>is a list of orders of representatives of conjugacy classes of the character table <var class="Arg">tbl</var>, in the same ordering as the conjugacy classes of <var class="Arg">tbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( tbl );</span>
-[ 1, 2, 3, 5, 5 ]
-</pre></div>
-
-<p><a id="X7CF7907F790A5DE6" name="X7CF7907F790A5DE6"></a></p>
-
-<h5>71.9-2 SizesCentralizers</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SizesCentralizers</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SizesCentralisers</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>is a list that stores at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-68-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-256" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-257" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-258" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-68" type="math/tex">i</script></span> the size of the centralizer of any element in the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-69-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-259" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-260" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-261" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-69" type="math/tex">i</script></span>-th conjugacy class of the character table <var class="Arg">tbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">SizesCentralizers( tbl );</span>
-[ 60, 4, 3, 5, 5 ]
-</pre></div>
-
-<p><a id="X7D9D2A45879A6A62" name="X7D9D2A45879A6A62"></a></p>
-
-<h5>71.9-3 SizesConjugacyClasses</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SizesConjugacyClasses</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>is a list that stores at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-70-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-262" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-263" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-264" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-70" type="math/tex">i</script></span> the size of the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-71-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-265" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-266" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-267" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-71" type="math/tex">i</script></span>-th conjugacy class of the character table <var class="Arg">tbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">SizesConjugacyClasses( tbl );</span>
-[ 1, 15, 20, 12, 12 ]
-</pre></div>
-
-<p><a id="X7C2753DE8094F4BA" name="X7C2753DE8094F4BA"></a></p>
-
-<h5>71.9-4 AutomorphismsOfTable</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ AutomorphismsOfTable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>is the permutation group of all column permutations of the character table <var class="Arg">tbl</var> that leave the set of irreducibles and each power map of <var class="Arg">tbl</var> invariant (see also <code class="func">TableAutomorphisms</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8082DD827C673138"><span class="RefLink">71.22-2</span></a>)).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "Dihedral", 8 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">AutomorphismsOfTable( tbl );</span>
-Group([ (4,5) ])
-<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( tbl );</span>
-[ 1, 4, 2, 2, 2 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">SizesConjugacyClasses( tbl );</span>
-[ 1, 2, 1, 2, 2 ]
-</pre></div>
-
-<p><a id="X7F58A82F7D88000A" name="X7F58A82F7D88000A"></a></p>
-
-<h5>71.9-5 <span class="Heading">UnderlyingCharacteristic</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ UnderlyingCharacteristic</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ UnderlyingCharacteristic</code>( <var class="Arg">psi</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For an ordinary character table <var class="Arg">tbl</var>, the result is <code class="code">0</code>, for a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-72-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-268" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-269" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-270" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-72" type="math/tex">p</script></span>-modular Brauer table <var class="Arg">tbl</var>, it is <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-73-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-271" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-272" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-273" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-73" type="math/tex">p</script></span>. The underlying characteristic of a class function <var class="Arg">psi</var> is equal to that of its underlying character table.</p>
-
-<p>The underlying characteristic must be stored when the table is constructed, there is no method to compute it.</p>
-
-<p>We cannot use the attribute <code class="func">Characteristic</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap31_mj.html#X81278E53800BF64D"><span class="RefLink">31.10-1</span></a>) to denote this, since of course each Brauer character is an element of characteristic zero in the sense of <strong class="pkg">GAP</strong> (see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X7C91D0D17850E564"><span class="RefLink">72</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingCharacteristic( tbl );</span>
-0
-<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingCharacteristic( tbl mod 17 );</span>
-17
-</pre></div>
-
-<p><a id="X804CFD597C795801" name="X804CFD597C795801"></a></p>
-
-<h5>71.9-6 <span class="Heading">Class Names and Character Names</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassNames</code>( <var class="Arg">tbl</var>[, <var class="Arg">"ATLAS"</var>] )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterNames</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p><code class="func">ClassNames</code> and <code class="func">CharacterNames</code> return lists of strings, one for each conjugacy class or irreducible character, respectively, of the character table <var class="Arg">tbl</var>. These names are used when <var class="Arg">tbl</var> is displayed.</p>
-
-<p>The default method for <code class="func">ClassNames</code> computes class names consisting of the order of an element in the class and at least one distinguishing letter.</p>
-
-<p>The default method for <code class="func">CharacterNames</code> returns the list <code class="code">[ "X.1", "X.2", ... ]</code>, whose length is the number of irreducible characters of <var class="Arg">tbl</var>.</p>
-
-<p>The position of the class with name <var class="Arg">name</var> in <var class="Arg">tbl</var> can be accessed as <code class="code"><var class="Arg">tbl</var>.<var class="Arg">name</var></code>.</p>
-
-<p>When <code class="func">ClassNames</code> is called with two arguments, the second being the string <code class="code">"ATLAS"</code>, the class names returned obey the convention used in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCCN85">[CCNPW85, Chapter 7, Section 5]</a>. If one is interested in "relative" class names of almost simple <strong class="pkg">Atlas</strong> groups, one can use the function <code class="func">AtlasClassNames</code> (<a href="http://www.gap-system.org/Manuals/pkg/atlasrep/doc/chap3.html#X78166D1D7D18EFBF"><span class="RefLink">AtlasRep: AtlasClassNames</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassNames( tbl );</span>
-[ "1a", "2a", "3a", "5a", "5b" ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbl.2a;</span>
-2
-</pre></div>
-
-<p><a id="X8333E8038308947E" name="X8333E8038308947E"></a></p>
-
-<h5>71.9-7 <span class="Heading">Class Parameters and Character Parameters</span></h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassParameters</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterParameters</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>The values of these attributes are lists containing a parameter for
-each conjugacy class or irreducible character, respectively, of the
-character table <var class="Arg">tbl</var>.</p>
-
-<p>It depends on <var class="Arg">tbl</var> what these parameters are, so there is no default to compute class and character parameters.</p>
-
-<p>For example, the classes of symmetric groups can be parametrized by
-partitions, corresponding to the cycle structures of permutations.
-Character tables constructed from generic character tables (see the
-manual of the <strong class="pkg">GAP</strong> Character Table Library) usually have class and character parameters stored.</p>
-
-<p>If <var class="Arg">tbl</var> is a <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-74-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-274" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-275" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-276" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-74" type="math/tex">p</script></span>-modular Brauer table such that class parameters are stored in the underlying ordinary table (see <code class="func">OrdinaryCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8011EEB684848039"><span class="RefLink">71.8-4</span></a>)) of <var class="Arg">tbl</var> then <code class="func">ClassParameters</code> returns the sublist of class parameters of the ordinary table, for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-75-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-277" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-278" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-279" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-75" type="math/tex">p</script></span>-regular classes.</p>
-
-<p><a id="X79C40EE97890202F" name="X79C40EE97890202F"></a></p>
-
-<h5>71.9-8 Identifier</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Identifier</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>is a string that identifies the character table <var class="Arg">tbl</var> in the current <strong class="pkg">GAP</strong> session. It is used mainly for class fusions into <var class="Arg">tbl</var>
- that are stored on other character tables. For character tables without
- group, the identifier is also used to print the table; this is the case
- for library tables, but also for tables that are constructed as direct
-products, factors etc. involving tables that may or may not store
-their groups.</p>
-
-<p>The default method for ordinary tables constructs strings of the form <code class="code">"CT<var class="Arg">n</var>"</code>, where <var class="Arg">n</var> is a positive integer. <code class="code">LARGEST_IDENTIFIER_NUMBER</code> is a list containing the largest integer <var class="Arg">n</var> used in the current <strong class="pkg">GAP</strong> session.</p>
-
-<p>The default method for Brauer tables returns the concatenation of the identifier of the ordinary table, the string <code class="code">"mod"</code>, and the (string of the) underlying characteristic.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">Identifier( CharacterTable( "A5" ) );</span>
-"A5"
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( Group( () ) );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Identifier( tbl ); Identifier( tbl mod 2 );</span>
-"CT8"
-"CT8mod2"
-</pre></div>
-
-<p><a id="X7932C35180C80953" name="X7932C35180C80953"></a></p>
-
-<h5>71.9-9 InfoText</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ InfoText</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( method )</td></tr></tbody></table></div>
-<p>is a mutable string with information about the character table <var class="Arg">tbl</var>. There is no default method to create an info text.</p>
-
-<p>This attribute is used mainly for library tables (see the manual of the <strong class="pkg">GAP</strong> Character Table Library). Usual parts of the information are the origin of the table, tests it has passed (<code class="code">1.o.r.</code> for the test of orthogonality, <code class="code">pow[<var class="Arg">p</var>]</code> for the construction of the <var class="Arg">p</var>-th power map, <code class="code">DEC</code> for the decomposition of ordinary into Brauer characters, <code class="code">TENS</code> for the decomposition of tensor products of irreducibles), and choices made without loss of generality.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">Print( InfoText( CharacterTable( "A5" ) ), "\n" );</span>
-origin: ATLAS of finite groups, tests: 1.o.r., pow[2,3,5]
-</pre></div>
-
-<p><a id="X7919E2897BE8234A" name="X7919E2897BE8234A"></a></p>
-
-<h5>71.9-10 InverseClasses</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ InverseClasses</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For a character table <var class="Arg">tbl</var>, <code class="func">InverseClasses</code> returns the list mapping each conjugacy class to its inverse class. This list can be regarded as <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-76-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.391em; display: inline-block;" role="math" id="MathJax-Span-280" class="math"><span style="display: inline-block; position: relative; width: 1.827em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1001.78em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-281" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-282" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-283" class="mo">−</span><span style="font-family: STIXGeneral;" id="MathJax-Span-284" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-285" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-76" type="math/tex">(-1)</script></span>-st power map of <var class="Arg">tbl</var> (see <code class="func">PowerMap</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>)).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">InverseClasses( CharacterTable( "A5" ) );</span>
-[ 1, 2, 3, 4, 5 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">InverseClasses( CharacterTable( "Cyclic", 3 ) );</span>
-[ 1, 3, 2 ]
-</pre></div>
-
-<p><a id="X87FF547981456932" name="X87FF547981456932"></a></p>
-
-<h5>71.9-11 RealClasses</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ RealClasses</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For a character table <var class="Arg">tbl</var>, <code class="func">RealClasses</code> returns the strictly sorted list of positions of classes in <var class="Arg">tbl</var> that consist of real elements.</p>
-
-<p>An element <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-77-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-286" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.43em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-287" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-288" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.713em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script id="MathJax-Element-77" type="math/tex">x</script></span> is <em>real</em> iff it is conjugate to its inverse <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></msup><mo>=</mo><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>o</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-78-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 6.828em; display: inline-block;" role="math" id="MathJax-Span-289" class="math"><span style="display: inline-block; position: relative; width: 5.24em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.659em, 1005.24em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-290" class="mrow"><span id="MathJax-Span-291" class="msubsup"><span style="display: inline-block; position: relative; width: 1.411em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-292" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.498em;"><span id="MathJax-Span-293" class="texatom"><span id="MathJax-Span-294" class="mrow"><span id="MathJax-Span-295" class="texatom"><span id="MathJax-Span-296" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-297" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-298" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-299" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-300" class="msubsup"><span style="display: inline-block; position: relative; width: 2.551em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-301" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.498em;"><span id="MathJax-Span-302" class="texatom"><span id="MathJax-Span-303" class="mrow"><span id="MathJax-Span-304" class="texatom"><span id="MathJax-Span-305" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-306" class="mi">o</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-307" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-308" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-309" class="mo">)</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-310" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-311" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.232em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></mrow></msup><mo>=</mo><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>o</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></mrow></msup></math></span></span><script id="MathJax-Element-78" type="math/tex">x^{{-1}} = x^{{o(x)-1}}</script></span>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">RealClasses( CharacterTable( "A5" ) );</span>
-[ 1, 2, 3, 4, 5 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">RealClasses( CharacterTable( "Cyclic", 3 ) );</span>
-[ 1 ]
-</pre></div>
-
-<p><a id="X7ABB007C799F7C49" name="X7ABB007C799F7C49"></a></p>
-
-<h5>71.9-12 ClassOrbit</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassOrbit</code>( <var class="Arg">tbl</var>, <var class="Arg">cc</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>is the list of positions of those conjugacy classes of the character table <var class="Arg">tbl</var> that are Galois conjugate to the <var class="Arg">cc</var>-th class. That is, exactly the classes at positions given by the list returned by <code class="func">ClassOrbit</code> contain generators of the cyclic group generated by an element in the <var class="Arg">cc</var>-th class.</p>
-
-<p>This information is computed from the power maps of <var class="Arg">tbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">ClassOrbit( CharacterTable( "A5" ), 4 );</span>
-[ 4, 5 ]
-</pre></div>
-
-<p><a id="X7F863B15804E0835" name="X7F863B15804E0835"></a></p>
-
-<h5>71.9-13 ClassRoots</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassRoots</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For a character table <var class="Arg">tbl</var>, <code class="func">ClassRoots</code> returns a list containing at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-79-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-312" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-313" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-314" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-79" type="math/tex">i</script></span> the list of positions of the classes of all nontrivial <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-80-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-315" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-316" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-317" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-80" type="math/tex">p</script></span>-th roots, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-81-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-318" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-319" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-320" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-81" type="math/tex">p</script></span> runs over the prime divisors of the <code class="func">Size</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X81EFD9FE804AC6EE"><span class="RefLink">71.8-5</span></a>) value of <var class="Arg">tbl</var>.</p>
-
-<p>This information is computed from the power maps of <var class="Arg">tbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">ClassRoots( CharacterTable( "A5" ) );</span>
-[ [ 2, 3, 4, 5 ], [ ], [ ], [ ], [ ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassRoots( CharacterTable( "Cyclic", 6 ) );</span>
-[ [ 3, 4, 5 ], [ ], [ 2 ], [ 2, 6 ], [ 6 ], [ ] ]
-</pre></div>
-
-<p><a id="X79CEBC3C7E0E63DF" name="X79CEBC3C7E0E63DF"></a></p>
-
-<h4>71.10 <span class="Heading">Normal Subgroups Represented by Lists of Class Positions</span></h4>
-
-<p>The following attributes for a character table <var class="Arg">tbl</var> correspond to attributes for the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-82-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-321" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-322" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-323" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-82" type="math/tex">G</script></span> of <var class="Arg">tbl</var>. But instead of a normal subgroup (or a list of normal subgroups) of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-83-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-324" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-325" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-326" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-83" type="math/tex">G</script></span>, they return a strictly sorted list of positive integers (or a list of such lists) which are the positions –relative to the <code class="func">ConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X849A38F887F6EC86"><span class="RefLink">71.6-2</span></a>) value of <var class="Arg">tbl</var>– of those classes forming the normal subgroup in question.</p>
-
-<p><a id="X850C7D947B3DBFA2" name="X850C7D947B3DBFA2"></a></p>
-
-<h5>71.10-1 ClassPositionsOfNormalSubgroups</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfNormalSubgroups</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfMaximalNormalSubgroups</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfMinimalNormalSubgroups</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>correspond to <code class="func">NormalSubgroups</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X80237A847E24E6CF"><span class="RefLink">39.19-8</span></a>), <code class="func">MaximalNormalSubgroups</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X82ECAA427C987318"><span class="RefLink">39.19-9</span></a>), <code class="func">MinimalNormalSubgroups</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X86FDD9BA819F5644"><span class="RefLink">39.19-10</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>
-
-<p>The entries of the result lists are sorted according to increasing
-length. (So this total order respects the partial order of normal
-subgroups given by inclusion.)</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfNormalSubgroups( tbls4 );</span>
-[ [ 1 ], [ 1, 3 ], [ 1, 3, 4 ], [ 1 .. 5 ] ]
-</pre></div>
-
-<p><a id="X8491DA0981D6F264" name="X8491DA0981D6F264"></a></p>
-
-<h5>71.10-2 ClassPositionsOfAgemo</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfAgemo</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>corresponds to <code class="func">Agemo</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X83DB33747F069ACC"><span class="RefLink">39.14-2</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfAgemo( tbls4, 2 );</span>
-[ 1, 3, 4 ]
-</pre></div>
-
-<p><a id="X7A6B1F8A84A495DC" name="X7A6B1F8A84A495DC"></a></p>
-
-<h5>71.10-3 ClassPositionsOfCentre</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfCentre</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfCenter</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>corresponds to <code class="func">Centre</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap35_mj.html#X847ABE6F781C7FE8"><span class="RefLink">35.4-5</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbld8:= CharacterTable( "Dihedral", 8 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfCentre( tbld8 );</span>
-[ 1, 3 ]
-</pre></div>
-
-<p><a id="X7D53F60785AB22B1" name="X7D53F60785AB22B1"></a></p>
-
-<h5>71.10-4 ClassPositionsOfDirectProductDecompositions</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfDirectProductDecompositions</code>( <var class="Arg">tbl</var>[, <var class="Arg">nclasses</var>] )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">tbl</var> be the ordinary character table of the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-84-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-327" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-328" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-329" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-84" type="math/tex">G</script></span>, say. Called with the only argument <var class="Arg">tbl</var>, <code class="func">ClassPositionsOfDirectProductDecompositions</code> returns the list of all those pairs <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>l</mi><mn>1</mn></msub><mo>,</mo><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-85-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.266em; display: inline-block;" role="math" id="MathJax-Span-330" class="math"><span style="display: inline-block; position: relative; width: 2.5em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1002.41em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-331" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-332" class="mo">[</span><span id="MathJax-Span-333" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-334" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-335" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-336" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-337" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-338" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-339" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-340" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.216em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>l</mi><mn>1</mn></msub><mo>,</mo><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-85" type="math/tex">[ l_1, l_2 ]</script></span> where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-86-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-341" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.72em, 2.746em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-342" class="mrow"><span id="MathJax-Span-343" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-344" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-345" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.208em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math></span></span><script id="MathJax-Element-86" type="math/tex">l_1</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-87-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-346" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.72em, 2.746em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-347" class="mrow"><span id="MathJax-Span-348" class="msubsup"><span style="display: inline-block; position: relative; width: 0.707em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.28em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-349" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.278em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-350" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.208em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math></span></span><script id="MathJax-Element-87" type="math/tex">l_2</script></span> are lists of class positions of normal subgroups <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mn>1</mn></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-88-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.453em; display: inline-block;" role="math" id="MathJax-Span-351" class="math"><span style="display: inline-block; position: relative; width: 1.106em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1001.11em, 2.746em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-352" class="mrow"><span id="MathJax-Span-353" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-354" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-355" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.169em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mn>1</mn></msub></math></span></span><script id="MathJax-Element-88" type="math/tex">N_1</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mn>2</mn></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-89-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.453em; display: inline-block;" role="math" id="MathJax-Span-356" class="math"><span style="display: inline-block; position: relative; width: 1.106em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1001.11em, 2.746em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-357" class="mrow"><span id="MathJax-Span-358" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-359" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-360" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.169em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mn>2</mn></msub></math></span></span><script id="MathJax-Element-89" type="math/tex">N_2</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-90-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-361" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-362" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-363" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-90" type="math/tex">G</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-91-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-364" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-365" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-366" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-91" type="math/tex">G</script></span> is their direct product and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>N</mi><mn>1</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>&#x2264;</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>N</mi><mn>2</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-92-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.578em; display: inline-block;" role="math" id="MathJax-Span-367" class="math"><span style="display: inline-block; position: relative; width: 4.279em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1004.21em, 2.938em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-368" class="mrow"><span id="MathJax-Span-369" class="texatom"><span id="MathJax-Span-370" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-371" class="mo">|</span></span></span><span id="MathJax-Span-372" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-373" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-374" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-375" class="texatom"><span id="MathJax-Span-376" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-377" class="mo">|</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-378" class="mo">≤</span><span style="padding-left: 0.313em;" id="MathJax-Span-379" class="texatom"><span id="MathJax-Span-380" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-381" class="mo">|</span></span></span><span id="MathJax-Span-382" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.73em, 4.15em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-383" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-384" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-385" class="texatom"><span id="MathJax-Span-386" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-387" class="mo">|</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.199em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>N</mi><mn>1</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>≤</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>N</mi><mn>2</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></math></span></span><script id="MathJax-Element-92" type="math/tex">|N_1| \leq |N_2|</script></span> holds. Called with second argument a list <var class="Arg">nclasses</var> of class positions of a normal subgroup <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-93-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-388" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-389" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-390" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-93" type="math/tex">N</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-94-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-391" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-392" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-393" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-94" type="math/tex">G</script></span>, <code class="func">ClassPositionsOfDirectProductDecompositions</code> returns the list of pairs describing the decomposition of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-95-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-394" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-395" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-396" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-95" type="math/tex">N</script></span> as a direct product of two normal subgroups of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-96-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-397" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-398" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-399" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-96" type="math/tex">G</script></span>.</p>
-
-<p><a id="X79EE7BE17BD343D5" name="X79EE7BE17BD343D5"></a></p>
-
-<h5>71.10-5 ClassPositionsOfDerivedSubgroup</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfDerivedSubgroup</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>corresponds to <code class="func">DerivedSubgroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X7CC17CF179ED7EF2"><span class="RefLink">39.12-3</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbld8:= CharacterTable( "Dihedral", 8 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfDerivedSubgroup( tbld8 );</span>
-[ 1, 3 ]
-</pre></div>
-
-<p><a id="X86ABB2E179D7F6E1" name="X86ABB2E179D7F6E1"></a></p>
-
-<h5>71.10-6 ClassPositionsOfElementaryAbelianSeries</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfElementaryAbelianSeries</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>corresponds to <code class="func">ElementaryAbelianSeries</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X83F057E5791944D6"><span class="RefLink">39.17-9</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbla5:= CharacterTable( "A5" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfElementaryAbelianSeries( tbls4 );</span>
-[ [ 1 .. 5 ], [ 1, 3, 4 ], [ 1, 3 ], [ 1 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfElementaryAbelianSeries( tbla5 );</span>
-fail
-</pre></div>
-
-<p><a id="X7D2A55A584F955DB" name="X7D2A55A584F955DB"></a></p>
-
-<h5>71.10-7 ClassPositionsOfFittingSubgroup</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfFittingSubgroup</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>corresponds to <code class="func">FittingSubgroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X780552B57C30DD8F"><span class="RefLink">39.12-5</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfFittingSubgroup( tbls4 );</span>
-[ 1, 3 ]
-</pre></div>
-
-<p><a id="X79AEFC4384769B72" name="X79AEFC4384769B72"></a></p>
-
-<h5>71.10-8 ClassPositionsOfLowerCentralSeries</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfLowerCentralSeries</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>corresponds to <code class="func">LowerCentralSeriesOfGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X879D55A67DB42676"><span class="RefLink">39.17-11</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbld8:= CharacterTable( "Dihedral", 8 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfLowerCentralSeries( tbls4 );</span>
-[ [ 1 .. 5 ], [ 1, 3, 4 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfLowerCentralSeries( tbld8 );</span>
-[ [ 1 .. 5 ], [ 1, 3 ], [ 1 ] ]
-</pre></div>
-
-<p><a id="X86065D217A36CD9B" name="X86065D217A36CD9B"></a></p>
-
-<h5>71.10-9 ClassPositionsOfUpperCentralSeries</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfUpperCentralSeries</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>corresponds to <code class="func">UpperCentralSeriesOfGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X8428592E8773CD7B"><span class="RefLink">39.17-12</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbld8:= CharacterTable( "Dihedral", 8 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfUpperCentralSeries( tbls4 );</span>
-[ [ 1 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfUpperCentralSeries( tbld8 );</span>
-[ [ 1, 3 ], [ 1, 2, 3, 4, 5 ] ]
-</pre></div>
-
-<p><a id="X8392DD5B813250A4" name="X8392DD5B813250A4"></a></p>
-
-<h5>71.10-10 ClassPositionsOfSupersolvableResiduum</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfSupersolvableResiduum</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>corresponds to <code class="func">SupersolvableResiduum</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X8440C61080CDAA14"><span class="RefLink">39.12-11</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfSupersolvableResiduum( tbls4 );</span>
-[ 1, 3 ]
-</pre></div>
-
-<p><a id="X7BBE7EBA7A64A6B0" name="X7BBE7EBA7A64A6B0"></a></p>
-
-<h5>71.10-11 ClassPositionsOfPCore</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfPCore</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>corresponds to <code class="func">PCore</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X7CF497C77B1E8938"><span class="RefLink">39.11-3</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfPCore( tbls4, 2 );</span>
-[ 1, 3 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfPCore( tbls4, 3 );</span>
-[ 1 ]
-</pre></div>
-
-<p><a id="X7FCF905D7FD7CC40" name="X7FCF905D7FD7CC40"></a></p>
-
-<h5>71.10-12 ClassPositionsOfNormalClosure</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfNormalClosure</code>( <var class="Arg">ordtbl</var>, <var class="Arg">classes</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>is the sorted list of the positions of all conjugacy classes of the ordinary character table <var class="Arg">ordtbl</var> that form the normal closure (see <code class="func">NormalClosure</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X7BDEA0A98720D1BB"><span class="RefLink">39.11-4</span></a>)) of the conjugacy classes at positions in the list <var class="Arg">classes</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfNormalClosure( tbls4, [ 1, 4 ] );</span>
-[ 1, 3, 4 ]
-</pre></div>
-
-<p><a id="X8733F0EA801785D4" name="X8733F0EA801785D4"></a></p>
-
-<h4>71.11 <span class="Heading">Operations Concerning Blocks</span></h4>
-
-<p><a id="X7ACB9306804F4E3F" name="X7ACB9306804F4E3F"></a></p>
-
-<h5>71.11-1 PrimeBlocks</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ PrimeBlocks</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ PrimeBlocksOp</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ComputedPrimeBlockss</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For an ordinary character table <var class="Arg">ordtbl</var> and a prime integer <var class="Arg">p</var>, <code class="func">PrimeBlocks</code> returns a record with the following components.</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code">block</code></strong></dt>
-<dd><p>a list, the value <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-97-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-400" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.848em, 1000.29em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-401" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-402" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.242em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></span></span><script id="MathJax-Element-97" type="math/tex">j</script></span> at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-98-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-403" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-404" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-405" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-98" type="math/tex">i</script></span> means that the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-99-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-406" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-407" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-408" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-99" type="math/tex">i</script></span>-th irreducible character of <var class="Arg">ordtbl</var> lies in the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-100-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-409" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.848em, 1000.29em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-410" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-411" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.242em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></span></span><script id="MathJax-Element-100" type="math/tex">j</script></span>-th <var class="Arg">p</var>-block of <var class="Arg">ordtbl</var>,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">defect</code></strong></dt>
-<dd><p>a list containing at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-101-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-412" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-413" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-414" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-101" type="math/tex">i</script></span> the defect of the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-102-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-415" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-416" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-417" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-102" type="math/tex">i</script></span>-th block,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">height</code></strong></dt>
-<dd><p>a list containing at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-103-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-418" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-419" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-420" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-103" type="math/tex">i</script></span> the height of the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-104-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-421" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-422" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-423" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-104" type="math/tex">i</script></span>-th irreducible character of <var class="Arg">ordtbl</var> in its block,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">relevant</code></strong></dt>
-<dd><p>a list of class positions such that only the restriction to these
- classes need be checked for deciding whether two characters lie in the
-same block, and</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">centralcharacter</code></strong></dt>
-<dd><p>a list containing at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-105-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-424" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-425" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-426" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-105" type="math/tex">i</script></span> a list whose values at the positions stored in the component <code class="code">relevant</code> are the values of a central character in the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-106-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-427" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-428" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-429" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-106" type="math/tex">i</script></span>-th block.</p>
-
-</dd>
-</dl>
-<p>The components <code class="code">relevant</code> and <code class="code">centralcharacters</code> are used by <code class="func">SameBlock</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E80E35985275F35"><span class="RefLink">71.11-2</span></a>).</p>
-
-<p>If <code class="func">InfoCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C6F3D947E5188D1"><span class="RefLink">71.4-2</span></a>) has level at least 2, the defects of the blocks and the heights of the characters are printed.</p>
-
-<p>The default method uses the attribute <code class="func">ComputedPrimeBlockss</code> for storing the computed value at position <var class="Arg">p</var>, and calls the operation <code class="func">PrimeBlocksOp</code> for computing values that are not yet known.</p>
-
-<p>Two ordinary irreducible characters <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C7;</mi><mo>,</mo><mi>&#x03C8;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-107-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.141em; display: inline-block;" role="math" id="MathJax-Span-430" class="math"><span style="display: inline-block; position: relative; width: 1.635em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1001.64em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-431" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-432" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-433" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-434" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 0.967em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>χ</mi><mo>,</mo><mi>ψ</mi></math></span></span><script id="MathJax-Element-107" type="math/tex">\chi, \psi</script></span> of a group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-108-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-435" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-436" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-437" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-108" type="math/tex">G</script></span> are said to lie in the same <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-109-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-438" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-439" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-440" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-109" type="math/tex">p</script></span>-<em>block</em> if the images of their central characters <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C9;</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03C7;</mi></mrow></msub><mo>,</mo><msub><mi>&#x03C9;</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03C8;</mi></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-110-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.641em; display: inline-block;" role="math" id="MathJax-Span-441" class="math"><span style="display: inline-block; position: relative; width: 2.788em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.061em, 1002.79em, 3.085em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-442" class="mrow"><span id="MathJax-Span-443" class="msubsup"><span style="display: inline-block; position: relative; width: 1.113em; height: 0px;"><span style="position: absolute; clip: rect(3.407em, 1000.65em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-444" class="mi">ω</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.686em;"><span id="MathJax-Span-445" class="texatom"><span id="MathJax-Span-446" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-447" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.029em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-448" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-449" class="msubsup"><span style="display: inline-block; position: relative; width: 1.233em; height: 0px;"><span style="position: absolute; clip: rect(3.407em, 1000.65em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-450" class="mi">ω</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.686em;"><span id="MathJax-Span-451" class="texatom"><span id="MathJax-Span-452" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-453" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.059em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.448em; border-left: 0px solid; width: 0px; height: 1.081em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow class="MJX-TeXAtom-ORD"><mi>χ</mi></mrow></msub><mo>,</mo><msub><mi>ω</mi><mrow class="MJX-TeXAtom-ORD"><mi>ψ</mi></mrow></msub></math></span></span><script id="MathJax-Element-110" type="math/tex">\omega_{\chi}, \omega_{\psi}</script></span> (see <code class="func">CentralCharacter</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X7DD8FDCF7FB7834A"><span class="RefLink">72.8-17</span></a>)) under the natural ring epimorphism <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">&#x2192;</mo><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>M</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-111-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.141em; display: inline-block;" role="math" id="MathJax-Span-454" class="math"><span style="display: inline-block; position: relative; width: 3.942em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1003.94em, 2.802em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-455" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-456" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-457" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-458" class="mi">R</span><span id="MathJax-Span-459" class="texatom"><span id="MathJax-Span-460" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-461" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-462" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.081em; border-left: 0px solid; width: 0px; height: 1.022em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>M</mi></math></span></span><script id="MathJax-Element-111" type="math/tex">R \rightarrow R / M</script></span> are equal, where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-112-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.828em; display: inline-block;" role="math" id="MathJax-Span-463" class="math"><span style="display: inline-block; position: relative; width: 0.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.6em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-464" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-465" class="mi">R</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></span></span><script id="MathJax-Element-112" type="math/tex">R</script></span> denotes the ring of algebraic integers in the complex number field, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-113-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.141em; display: inline-block;" role="math" id="MathJax-Span-466" class="math"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.86em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-467" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-468" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script id="MathJax-Element-113" type="math/tex">M</script></span> is a maximal ideal in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-114-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.828em; display: inline-block;" role="math" id="MathJax-Span-469" class="math"><span style="display: inline-block; position: relative; width: 0.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.6em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-470" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-471" class="mi">R</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></span></span><script id="MathJax-Element-114" type="math/tex">R</script></span> with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>R</mi><mo>&#x2286;</mo><mi>M</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-115-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.266em; display: inline-block;" role="math" id="MathJax-Span-472" class="math"><span style="display: inline-block; position: relative; width: 3.269em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1003.27em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-473" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-474" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-475" class="mi">R</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-476" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-477" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.24em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>R</mi><mo>⊆</mo><mi>M</mi></math></span></span><script id="MathJax-Element-115" type="math/tex">pR \subseteq M</script></span>. (The distribution to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-116-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-478" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-479" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-480" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-116" type="math/tex">p</script></span>-blocks is in fact independent of the choice of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-117-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.141em; display: inline-block;" role="math" id="MathJax-Span-481" class="math"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.86em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-482" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-483" class="mi">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.039em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script id="MathJax-Element-117" type="math/tex">M</script></span>, see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBIsa76">[Isa76]</a>.)</p>
-
-<p>For <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mi>p</mi><mi>a</mi></msup><mi>m</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-118-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.266em; display: inline-block;" role="math" id="MathJax-Span-484" class="math"><span style="display: inline-block; position: relative; width: 4.038em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1004.02em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-485" class="mrow"><span id="MathJax-Span-486" class="texatom"><span id="MathJax-Span-487" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-488" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-489" class="mi">G</span><span id="MathJax-Span-490" class="texatom"><span id="MathJax-Span-491" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-492" class="mo">|</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-493" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-494" class="msubsup"><span style="display: inline-block; position: relative; width: 0.933em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.34em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-495" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.504em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-496" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-497" class="mi">m</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.27em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><msup><mi>p</mi><mi>a</mi></msup><mi>m</mi></math></span></span><script id="MathJax-Element-118" type="math/tex">|G| = p^a m</script></span> where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-119-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-498" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-499" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-500" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-119" type="math/tex">p</script></span> does not divide <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-120-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-501" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.7em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-502" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-503" class="mi">m</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script id="MathJax-Element-120" type="math/tex">m</script></span>, the <em>defect</em> of a block is the integer <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-121-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-504" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.53em, 2.801em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-505" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-506" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.079em; border-left: 0px solid; width: 0px; height: 1.03em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script id="MathJax-Element-121" type="math/tex">d</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>&#x2212;</mo><mi>d</mi></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-122-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.328em; display: inline-block;" role="math" id="MathJax-Span-507" class="math"><span style="display: inline-block; position: relative; width: 1.779em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.462em, 1001.78em, 2.801em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-508" class="mrow"><span id="MathJax-Span-509" class="msubsup"><span style="display: inline-block; position: relative; width: 1.79em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.34em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-510" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.504em;"><span id="MathJax-Span-511" class="texatom"><span id="MathJax-Span-512" class="mrow"><span id="MathJax-Span-513" class="texatom"><span id="MathJax-Span-514" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-515" class="mi">a</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-516" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-517" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.491em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>−</mo><mi>d</mi></mrow></mrow></msup></math></span></span><script id="MathJax-Element-122" type="math/tex">p^{{a-d}}</script></span> is the largest power of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-123-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-518" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-519" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-520" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-123" type="math/tex">p</script></span> that divides the degrees of all characters in the block.</p>
-
-<p>The <em>height</em> of a character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C7;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-124-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-521" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.53em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-522" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-523" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 0.967em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>χ</mi></math></span></span><script id="MathJax-Element-124" type="math/tex">\chi</script></span> in the block is defined as the largest exponent <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-125-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-524" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-525" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-526" class="mi">h</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 1.025em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></span></span><script id="MathJax-Element-125" type="math/tex">h</script></span> for which <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mi>h</mi></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-126-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.203em; display: inline-block;" role="math" id="MathJax-Span-527" class="math"><span style="display: inline-block; position: relative; width: 0.913em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.462em, 1000.91em, 2.801em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-528" class="mrow"><span id="MathJax-Span-529" class="msubsup"><span style="display: inline-block; position: relative; width: 0.933em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.34em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-530" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.504em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-531" class="mi">h</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.491em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mi>h</mi></msup></math></span></span><script id="MathJax-Element-126" type="math/tex">p^h</script></span> divides <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C7;</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>&#x2212;</mo><mi>d</mi></mrow></mrow></msup></math>" style="position: relative;" tabindex="0" id="MathJax-Element-127-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.891em; display: inline-block;" role="math" id="MathJax-Span-532" class="math"><span style="display: inline-block; position: relative; width: 3.75em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.654em, 1003.75em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-533" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-534" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-535" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-536" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-537" class="mo">)</span><span id="MathJax-Span-538" class="texatom"><span id="MathJax-Span-539" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-540" class="mo">/</span></span></span><span id="MathJax-Span-541" class="msubsup"><span style="display: inline-block; position: relative; width: 1.79em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.34em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-542" class="mi">p</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.504em;"><span id="MathJax-Span-543" class="texatom"><span id="MathJax-Span-544" class="mrow"><span id="MathJax-Span-545" class="texatom"><span id="MathJax-Span-546" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-547" class="mi">a</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-548" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-549" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.494em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>χ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>−</mo><mi>d</mi></mrow></mrow></msup></math></span></span><script id="MathJax-Element-127" type="math/tex">\chi(1) / p^{{a-d}}</script></span>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "L3(2)" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">pbl:= PrimeBlocks( tbl, 2 );</span>
-rec( block := [ 1, 1, 1, 1, 1, 2 ],
- centralcharacter := [ [ ,, 56,, 24 ], [ ,, -7,, 3 ] ],
- defect := [ 3, 0 ], height := [ 0, 0, 0, 1, 0, 0 ],
- relevant := [ 3, 5 ] )
-</pre></div>
-
-<p><a id="X7E80E35985275F35" name="X7E80E35985275F35"></a></p>
-
-<h5>71.11-2 SameBlock</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SameBlock</code>( <var class="Arg">p</var>, <var class="Arg">omega1</var>, <var class="Arg">omega2</var>, <var class="Arg">relevant</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">p</var> be a prime integer, <var class="Arg">omega1</var> and <var class="Arg">omega2</var> be two central characters (or their values lists) of a character table, and <var class="Arg">relevant</var> be a list of positions as is stored in the component <code class="code">relevant</code> of a record returned by <code class="func">PrimeBlocks</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7ACB9306804F4E3F"><span class="RefLink">71.11-1</span></a>).</p>
-
-<p><code class="func">SameBlock</code> returns <code class="keyw">true</code> if <var class="Arg">omega1</var> and <var class="Arg">omega2</var> are equal modulo any maximal ideal in the ring of complex algebraic integers containing the ideal spanned by <var class="Arg">p</var>, and <code class="keyw">false</code> otherwise.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">omega:= List( Irr( tbl ), CentralCharacter );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">SameBlock( 2, omega[1], omega[2], pbl.relevant );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">SameBlock( 2, omega[1], omega[6], pbl.relevant );</span>
-false
-</pre></div>
-
-<p><a id="X7FF4CE4A7A272F88" name="X7FF4CE4A7A272F88"></a></p>
-
-<h5>71.11-3 BlocksInfo</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ BlocksInfo</code>( <var class="Arg">modtbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For a Brauer character table <var class="Arg">modtbl</var>, the value of <code class="func">BlocksInfo</code> is a list of (mutable) records, the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-128-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-550" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-551" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-552" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-128" type="math/tex">i</script></span>-th entry containing information about the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-129-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-553" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-554" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-555" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-129" type="math/tex">i</script></span>-th block. Each record has the following components.</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code">defect</code></strong></dt>
-<dd><p>the defect of the block,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">ordchars</code></strong></dt>
-<dd><p>the list of positions of the ordinary characters that belong to the block, relative to <code class="code">Irr( OrdinaryCharacterTable( <var class="Arg">modtbl</var> ) )</code>,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">modchars</code></strong></dt>
-<dd><p>the list of positions of the Brauer characters that belong to the block, relative to <code class="code">IBr( <var class="Arg">modtbl</var> )</code>.</p>
-
-</dd>
-</dl>
-<p>Optional components are</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code">basicset</code></strong></dt>
-<dd><p>a list of positions of ordinary characters in the block whose restriction to <var class="Arg">modtbl</var> is maximally linearly independent, relative to <code class="code">Irr( OrdinaryCharacterTable( <var class="Arg">modtbl</var> ) )</code>,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">decmat</code></strong></dt>
-<dd><p>the decomposition matrix of the block, it is stored automatically when <code class="func">DecompositionMatrix</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X84701640811D2345"><span class="RefLink">71.11-4</span></a>) is called for the block,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">decinv</code></strong></dt>
-<dd><p>inverse of the decomposition matrix of the block, restricted to the ordinary characters described by <code class="code">basicset</code>,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">brauertree</code></strong></dt>
-<dd><p>a list that describes the Brauer tree of the block, in the case that the block is of defect <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-130-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-556" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.38em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-557" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-558" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script id="MathJax-Element-130" type="math/tex">1</script></span>.</p>
-
-</dd>
-</dl>
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">BlocksInfo( CharacterTable( "L3(2)" ) mod 2 );</span>
-[ rec( basicset := [ 1, 2, 3 ],
- decinv := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
- defect := 3, modchars := [ 1, 2, 3 ],
- ordchars := [ 1, 2, 3, 4, 5 ] ),
- rec( basicset := [ 6 ], decinv := [ [ 1 ] ], defect := 0,
- modchars := [ 4 ], ordchars := [ 6 ] ) ]
-</pre></div>
-
-<p><a id="X84701640811D2345" name="X84701640811D2345"></a></p>
-
-<h5>71.11-4 DecompositionMatrix</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ DecompositionMatrix</code>( <var class="Arg">modtbl</var>[, <var class="Arg">blocknr</var>] )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">modtbl</var> be a Brauer character table.</p>
-
-<p>Called with one argument, <code class="func">DecompositionMatrix</code> returns the decomposition matrix of <var class="Arg">modtbl</var>, where the rows and columns are indexed by the irreducible characters of the ordinary character table of <var class="Arg">modtbl</var> and the irreducible characters of <var class="Arg">modtbl</var>, respectively,</p>
-
-<p>Called with two arguments, <code class="func">DecompositionMatrix</code> returns the decomposition matrix of the block of <var class="Arg">modtbl</var> with number <var class="Arg">blocknr</var>; the matrix is stored as value of the <code class="code">decmat</code> component of the <var class="Arg">blocknr</var>-th entry of the <code class="func">BlocksInfo</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FF4CE4A7A272F88"><span class="RefLink">71.11-3</span></a>) list of <var class="Arg">modtbl</var>.</p>
-
-<p>An ordinary irreducible character is in block <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-131-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-559" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-560" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-561" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-131" type="math/tex">i</script></span> if and only if all characters before the first character of the same block lie in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>&#x2212;</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-132-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.516em; display: inline-block;" role="math" id="MathJax-Span-562" class="math"><span style="display: inline-block; position: relative; width: 1.923em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1001.82em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-563" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-564" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-565" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-566" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.018em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>−</mo><mn>1</mn></math></span></span><script id="MathJax-Element-132" type="math/tex">i-1</script></span> different blocks. An irreducible Brauer character is in block <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-133-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-567" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-568" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-569" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-133" type="math/tex">i</script></span> if it has nonzero scalar product with an ordinary irreducible character in block <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-134-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-570" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-571" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-572" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-134" type="math/tex">i</script></span>.</p>
-
-<p><code class="func">DecompositionMatrix</code> is based on the more general function <code class="func">Decomposition</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap25_mj.html#X7911A60384C511AB"><span class="RefLink">25.4-1</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">modtbl:= CharacterTable( "L3(2)" ) mod 2;</span>
-BrauerTable( "L3(2)", 2 )
-<span class="GAPprompt">gap></span> <span class="GAPinput">DecompositionMatrix( modtbl );</span>
-[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 1, 0 ],
- [ 1, 1, 1, 0 ], [ 0, 0, 0, 1 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">DecompositionMatrix( modtbl, 1 );</span>
-[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ], [ 0, 1, 1 ], [ 1, 1, 1 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">DecompositionMatrix( modtbl, 2 );</span>
-[ [ 1 ] ]
-</pre></div>
-
-<p><a id="X83EC921380AF9B3B" name="X83EC921380AF9B3B"></a></p>
-
-<h5>71.11-5 LaTeXStringDecompositionMatrix</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ LaTeXStringDecompositionMatrix</code>( <var class="Arg">modtbl</var>[, <var class="Arg">blocknr</var>][, <var class="Arg">options</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>is a string that contains LaTeX code to print a decomposition matrix (see <code class="func">DecompositionMatrix</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X84701640811D2345"><span class="RefLink">71.11-4</span></a>)) nicely.</p>
-
-<p>The optional argument <var class="Arg">options</var>, if present, must be a record with components <code class="code">phi</code>, <code class="code">chi</code> (strings used in each label for columns and rows), <code class="code">collabels</code>, <code class="code">rowlabels</code> (subscripts for the labels). The defaults for <code class="code">phi</code> and <code class="code">chi</code> are <code class="code">"{\\tt Y}"</code> and <code class="code">"{\\tt X}"</code>, the defaults for <code class="code">collabels</code> and <code class="code">rowlabels</code>
- are the lists of positions of the Brauer characters and ordinary
-characters in the respective lists of irreducibles in the character
-tables.</p>
-
-<p>The optional components <code class="code">nrows</code> and <code class="code">ncols</code> denote the maximal number of rows and columns per array; if they are present then each portion of <code class="code">nrows</code> rows and <code class="code">ncols</code> columns forms an array of its own which is enclosed in <code class="code">\[</code>, <code class="code">\]</code>.</p>
-
-<p>If the component <code class="code">decmat</code> is bound in <var class="Arg">options</var> then it must be the decomposition matrix in question, in this case the matrix is not computed from the information in <var class="Arg">modtbl</var>.</p>
-
-<p>For those character tables from the <strong class="pkg">GAP</strong> table library that belong to the <strong class="pkg">Atlas</strong> of Finite Groups <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCCN85">[CCNPW85]</a>, <code class="func">AtlasLabelsOfIrreducibles</code> (<a href="http://www.gap-system.org/Manuals/pkg/ctbllib/doc/chap4.html#X7ADC9DC980CF0685"><span class="RefLink">CTblLib: AtlasLabelsOfIrreducibles</span></a>) constructs character labels that are compatible with those used in the <strong class="pkg">Atlas</strong> (see <a href="http://www.gap-system.org/Manuals/pkg/ctbllib/doc/chap4.html#X81FE89697EAAB1EA"><span class="RefLink">CTblLib: Atlas Tables</span></a> in the manual of the <strong class="pkg">GAP</strong> Character Table Library).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">modtbl:= CharacterTable( "L3(2)" ) mod 2;;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Print( LaTeXStringDecompositionMatrix( modtbl, 1 ) );</span>
-\[
-\begin{array}{r|rrr} \hline
- & {\tt Y}_{1}
- & {\tt Y}_{2}
- & {\tt Y}_{3}
- \rule[-7pt]{0pt}{20pt} \\ \hline
-{\tt X}_{1} & 1 & . & . \rule[0pt]{0pt}{13pt} \\
-{\tt X}_{2} & . & 1 & . \\
-{\tt X}_{3} & . & . & 1 \\
-{\tt X}_{4} & . & 1 & 1 \\
-{\tt X}_{5} & 1 & 1 & 1 \rule[-7pt]{0pt}{5pt} \\
-\hline
-\end{array}
-\]
-<span class="GAPprompt">gap></span> <span class="GAPinput">options:= rec( phi:= "\\varphi", chi:= "\\chi" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Print( LaTeXStringDecompositionMatrix( modtbl, 1, options ) );</span>
-\[
-\begin{array}{r|rrr} \hline
- & \varphi_{1}
- & \varphi_{2}
- & \varphi_{3}
- \rule[-7pt]{0pt}{20pt} \\ \hline
-\chi_{1} & 1 & . & . \rule[0pt]{0pt}{13pt} \\
-\chi_{2} & . & 1 & . \\
-\chi_{3} & . & . & 1 \\
-\chi_{4} & . & 1 & 1 \\
-\chi_{5} & 1 & 1 & 1 \rule[-7pt]{0pt}{5pt} \\
-\hline
-\end{array}
-\]
-</pre></div>
-
-<p><a id="X873211618402ACF7" name="X873211618402ACF7"></a></p>
-
-<h4>71.12 <span class="Heading">Other Operations for Character Tables</span></h4>
-
-<p>In the following, we list operations for character tables that are not attributes.</p>
-
-<p><a id="X8441983C845F2176" name="X8441983C845F2176"></a></p>
-
-<h5>71.12-1 Index</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Index</code>( <var class="Arg">tbl</var>, <var class="Arg">subtbl</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>For two character tables <var class="Arg">tbl</var> and <var class="Arg">subtbl</var>, <code class="func">Index</code> returns the quotient of the <code class="func">Size</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X81EFD9FE804AC6EE"><span class="RefLink">71.8-5</span></a>) values of <var class="Arg">tbl</var> and <var class="Arg">subtbl</var>. The containment of the underlying groups of <var class="Arg">subtbl</var> and <var class="Arg">tbl</var> is <em>not</em> checked; so the distinction between <code class="func">Index</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X86DC933779B1CABD"><span class="RefLink">39.3-2</span></a>) and <code class="func">IndexNC</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X86DC933779B1CABD"><span class="RefLink">39.3-2</span></a>) is not made for character tables.</p>
-
-<p><a id="X8123650E817926FC" name="X8123650E817926FC"></a></p>
-
-<h5>71.12-2 IsInternallyConsistent</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsInternallyConsistent</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( method )</td></tr></tbody></table></div>
-<p>For an <em>ordinary</em> character table <var class="Arg">tbl</var>, <code class="func">IsInternallyConsistent</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap12_mj.html#X7F6C5C3287E8B816"><span class="RefLink">12.8-4</span></a>) checks the consistency of the following attribute values (if stored).</p>
-
-
-<ul>
-<li><p><code class="func">Size</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap30_mj.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>), <code class="func">SizesCentralizers</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CF7907F790A5DE6"><span class="RefLink">71.9-2</span></a>), and <code class="func">SizesConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D9D2A45879A6A62"><span class="RefLink">71.9-3</span></a>).</p>
-
-</li>
-<li><p><code class="func">SizesCentralizers</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CF7907F790A5DE6"><span class="RefLink">71.9-2</span></a>) and <code class="func">OrdersClassRepresentatives</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>).</p>
-
-</li>
-<li><p><code class="func">ComputedPowerMaps</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>) and <code class="func">OrdersClassRepresentatives</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>).</p>
-
-</li>
-<li><p><code class="func">SizesCentralizers</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CF7907F790A5DE6"><span class="RefLink">71.9-2</span></a>) and <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>).</p>
-
-</li>
-<li><p><code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) (first orthogonality relation).</p>
-
-</li>
-</ul>
-<p>For a <em>Brauer</em> table <var class="Arg">tbl</var>, <code class="func">IsInternallyConsistent</code> checks the consistency of the following attribute values (if stored).</p>
-
-
-<ul>
-<li><p><code class="func">Size</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap30_mj.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>), <code class="func">SizesCentralizers</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CF7907F790A5DE6"><span class="RefLink">71.9-2</span></a>), and <code class="func">SizesConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D9D2A45879A6A62"><span class="RefLink">71.9-3</span></a>).</p>
-
-</li>
-<li><p><code class="func">SizesCentralizers</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CF7907F790A5DE6"><span class="RefLink">71.9-2</span></a>) and <code class="func">OrdersClassRepresentatives</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>).</p>
-
-</li>
-<li><p><code class="func">ComputedPowerMaps</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>) and <code class="func">OrdersClassRepresentatives</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>).</p>
-
-</li>
-<li><p><code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) (closure under complex conjugation and Frobenius map).</p>
-
-</li>
-</ul>
-<p>If no inconsistency occurs, <code class="keyw">true</code> is returned, otherwise each inconsistency is printed to the screen if the level of <code class="func">InfoWarning</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap7_mj.html#X7A28F77C82D6A3E0"><span class="RefLink">7.4-7</span></a>) is at least <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-135-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-573" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.38em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-574" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-575" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script id="MathJax-Element-135" type="math/tex">1</script></span> (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap7_mj.html#X7A9C902479CB6F7C"><span class="RefLink">7.4</span></a>), and <code class="keyw">false</code> is returned at the end.</p>
-
-<p><a id="X7A0CBD1884276882" name="X7A0CBD1884276882"></a></p>
-
-<h5>71.12-3 IsPSolvableCharacterTable</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsPSolvableCharacterTable</code>( <var class="Arg">tbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsPSolubleCharacterTable</code>( <var class="Arg">tbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsPSolvableCharacterTableOp</code>( <var class="Arg">tbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsPSolubleCharacterTableOp</code>( <var class="Arg">tbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ComputedIsPSolvableCharacterTables</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ComputedIsPSolubleCharacterTables</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p><code class="func">IsPSolvableCharacterTable</code> for the ordinary character table <var class="Arg">tbl</var> corresponds to <code class="func">IsPSolvable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html#X81130F9A7CFCF6BF"><span class="RefLink">39.15-23</span></a>) for the group of <var class="Arg">tbl</var>, <var class="Arg">p</var> must be either a prime integer or <code class="code">0</code>.</p>
-
-<p>The default method uses the attribute <code class="func">ComputedIsPSolvableCharacterTables</code> for storing the computed value at position <var class="Arg">p</var>, and calls the operation <code class="func">IsPSolvableCharacterTableOp</code> for computing values that are not yet known.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "Sz(8)" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsPSolvableCharacterTable( tbl, 2 );</span>
-false
-<span class="GAPprompt">gap></span> <span class="GAPinput">IsPSolvableCharacterTable( tbl, 3 );</span>
-true
-</pre></div>
-
-<p><a id="X82F523E8784B3752" name="X82F523E8784B3752"></a></p>
-
-<h5>71.12-4 IsClassFusionOfNormalSubgroup</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsClassFusionOfNormalSubgroup</code>( <var class="Arg">subtbl</var>, <var class="Arg">fus</var>, <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>For two ordinary character tables <var class="Arg">tbl</var> and <var class="Arg">subtbl</var> of a group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-136-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-576" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-577" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-578" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-136" type="math/tex">G</script></span> and its subgroup <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-137-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-579" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.77em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-580" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-581" class="mi">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.043em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.997em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script id="MathJax-Element-137" type="math/tex">U</script></span>, say, and a list <var class="Arg">fus</var> of positive integers that describes the class fusion of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-138-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-582" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.77em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-583" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-584" class="mi">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.043em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.997em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script id="MathJax-Element-138" type="math/tex">U</script></span> into <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-139-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-585" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-586" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-587" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-139" type="math/tex">G</script></span>, <code class="func">IsClassFusionOfNormalSubgroup</code> returns <code class="keyw">true</code> if <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-140-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-588" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.77em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-589" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-590" class="mi">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.043em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.997em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script id="MathJax-Element-140" type="math/tex">U</script></span> is a normal subgroup of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-141-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-591" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-592" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-593" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-141" type="math/tex">G</script></span>, and <code class="keyw">false</code> otherwise.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tblc2:= CharacterTable( "Cyclic", 2 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbld8:= CharacterTable( "Dihedral", 8 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleClassFusions( tblc2, tbld8 );</span>
-[ [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List(fus, map -> IsClassFusionOfNormalSubgroup(tblc2, map, tbld8));</span>
-[ true, false, false ]
-</pre></div>
-
-<p><a id="X7FD3D3047DE6381E" name="X7FD3D3047DE6381E"></a></p>
-
-<h5>71.12-5 Indicator</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Indicator</code>( <var class="Arg">tbl</var>[, <var class="Arg">characters</var>], <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IndicatorOp</code>( <var class="Arg">tbl</var>, <var class="Arg">characters</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ComputedIndicators</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>If <var class="Arg">tbl</var> is an ordinary character table then <code class="func">Indicator</code> returns the list of <var class="Arg">n</var>-th Frobenius-Schur indicators of the characters in the list <var class="Arg">characters</var>; the default of <var class="Arg">characters</var> is <code class="code">Irr( <var class="Arg">tbl</var> )</code>.</p>
-
-<p>The <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-142-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-594" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-595" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-596" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-142" type="math/tex">n</script></span>-th Frobenius-Schur indicator <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03BD;</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>&#x03C7;</mi><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-143-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.766em; display: inline-block;" role="math" id="MathJax-Span-597" class="math"><span style="display: inline-block; position: relative; width: 2.115em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.07em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-598" class="mrow"><span id="MathJax-Span-599" class="msubsup"><span style="display: inline-block; position: relative; width: 0.899em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.46em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-600" class="mi">ν</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.47em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-601" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-602" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-603" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-604" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.273em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ν</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-143" type="math/tex">\nu_n(\chi)</script></span> of an ordinary character <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C7;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-144-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-605" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.53em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-606" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-607" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 0.967em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>χ</mi></math></span></span><script id="MathJax-Element-144" type="math/tex">\chi</script></span> of the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-145-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-608" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-609" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-610" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-145" type="math/tex">G</script></span> is given by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03BD;</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>&#x03C7;</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><munder><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>g</mi><mo>&#x2208;</mo><mi>G</mi></mrow></mrow></munder><mi>&#x03C7;</mi><mo stretchy="false">(</mo><msup><mi>g</mi><mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></math>" style="position: relative;" tabindex="0" id="MathJax-Element-146-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 13.141em; display: inline-block;" role="math" id="MathJax-Span-611" class="math"><span style="display: inline-block; position: relative; width: 10.096em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.739em, 1010.03em, 3.23em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-612" class="mrow"><span id="MathJax-Span-613" class="msubsup"><span style="display: inline-block; position: relative; width: 0.899em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.46em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-614" class="mi">ν</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.47em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-615" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-616" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-617" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-618" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-619" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-620" class="mo">(</span><span id="MathJax-Span-621" class="munderover"><span style="display: inline-block; position: relative; width: 2.337em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-622" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.694em; left: 0.914em;"><span id="MathJax-Span-623" class="texatom"><span id="MathJax-Span-624" class="mrow"><span id="MathJax-Span-625" class="texatom"><span id="MathJax-Span-626" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-627" class="mi">g</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-628" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-629" class="mi">G</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-630" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-631" class="mo">(</span><span id="MathJax-Span-632" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-633" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-634" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-635" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-636" class="mo">)</span><span id="MathJax-Span-637" class="texatom"><span id="MathJax-Span-638" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-639" class="mo">/</span></span></span><span id="MathJax-Span-640" class="texatom"><span id="MathJax-Span-641" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-642" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-643" class="mi">G</span><span id="MathJax-Span-644" class="texatom"><span id="MathJax-Span-645" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-646" class="mo">|</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.637em; border-left: 0px solid; width: 0px; height: 1.689em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ν</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></mrow></munder><mi>χ</mi><mo stretchy="false">(</mo><msup><mi>g</mi><mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></math></span></span><script id="MathJax-Element-146" type="math/tex">\nu_n(\chi) = ( \sum_{{g \in G}} \chi(g^n) ) / |G|</script></span>.</p>
-
-<p>If <var class="Arg">tbl</var> is a Brauer table in characteristic <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2260;</mo><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-147-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.953em; display: inline-block;" role="math" id="MathJax-Span-647" class="math"><span style="display: inline-block; position: relative; width: 1.49em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1001.46em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-648" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-649" class="mo">≠</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-650" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.207em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≠</mo><mn>2</mn></math></span></span><script id="MathJax-Element-147" type="math/tex"> \neq 2</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">n</mtext></mrow><mo>=</mo><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-148-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.953em; display: inline-block;" role="math" id="MathJax-Span-651" class="math"><span style="display: inline-block; position: relative; width: 2.26em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.23em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-652" class="mrow"><span id="MathJax-Span-653" class="texatom"><span id="MathJax-Span-654" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-655" class="mtext">n</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-656" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-657" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 1.016em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">n</mtext></mrow><mo>=</mo><mn>2</mn></math></span></span><script id="MathJax-Element-148" type="math/tex">\textit{n} = 2</script></span> then <code class="func">Indicator</code> returns the second indicator.</p>
-
-<p>The default method uses the attribute <code class="func">ComputedIndicators</code> for storing the computed value at position <var class="Arg">n</var>, and calls the operation <code class="func">IndicatorOp</code> for computing values that are not yet known.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "L3(2)" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Indicator( tbl, 2 );</span>
-[ 1, 0, 0, 1, 1, 1 ]
-</pre></div>
-
-<p><a id="X83AE05BF8085B3C8" name="X83AE05BF8085B3C8"></a></p>
-
-<h5>71.12-6 NrPolyhedralSubgroups</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ NrPolyhedralSubgroups</code>( <var class="Arg">tbl</var>, <var class="Arg">c1</var>, <var class="Arg">c2</var>, <var class="Arg">c3</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>returns the number and isomorphism type of polyhedral subgroups of the group with ordinary character table <var class="Arg">tbl</var> which are generated by an element <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-149-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-658" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.994em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-659" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-660" class="mi">g</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 0.966em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></span></span><script id="MathJax-Element-149" type="math/tex">g</script></span> of class <var class="Arg">c1</var> and an element <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-150-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-661" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-662" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-663" class="mi">h</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 1.025em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></span></span><script id="MathJax-Element-150" type="math/tex">h</script></span> of class <var class="Arg">c2</var> with the property that the product <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>h</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-151-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-664" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1000.94em, 2.994em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-665" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-666" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-667" class="mi">h</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 1.281em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>h</mi></math></span></span><script id="MathJax-Element-151" type="math/tex">gh</script></span> lies in class <var class="Arg">c3</var>.</p>
-
-<p>According to <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBNPP84">[NPP84, p. 233]</a>, the number of polyhedral subgroups of isomorphism type <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>4</mn></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-152-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-668" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1001.06em, 2.746em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-669" class="mrow"><span id="MathJax-Span-670" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-671" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-672" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.169em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>4</mn></msub></math></span></span><script id="MathJax-Element-152" type="math/tex">V_4</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>n</mi></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-153-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.953em; display: inline-block;" role="math" id="MathJax-Span-673" class="math"><span style="display: inline-block; position: relative; width: 1.49em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1001.49em, 2.753em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-674" class="mrow"><span id="MathJax-Span-675" class="msubsup"><span style="display: inline-block; position: relative; width: 1.504em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.7em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-676" class="mi">D</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.722em;"><span id="MathJax-Span-677" class="texatom"><span id="MathJax-Span-678" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-679" class="mn">2</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-680" class="mi">n</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.177em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>n</mi></mrow></msub></math></span></span><script id="MathJax-Element-153" type="math/tex">D_{2n}</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mn>4</mn></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-154-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-681" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.64em, 1001.06em, 2.746em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-682" class="mrow"><span id="MathJax-Span-683" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-684" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-685" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.188em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mn>4</mn></msub></math></span></span><script id="MathJax-Element-154" type="math/tex">A_4</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>4</mn></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-155-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.203em; display: inline-block;" role="math" id="MathJax-Span-686" class="math"><span style="display: inline-block; position: relative; width: 0.913em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.641em, 1000.91em, 2.746em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-687" class="mrow"><span id="MathJax-Span-688" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.179em, 1000.51em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-689" class="mi">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-690" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.187em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>4</mn></msub></math></span></span><script id="MathJax-Element-155" type="math/tex">S_4</script></span>, and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mn>5</mn></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-156-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.391em; display: inline-block;" role="math" id="MathJax-Span-691" class="math"><span style="display: inline-block; position: relative; width: 1.058em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.64em, 1001.06em, 2.756em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-692" class="mrow"><span id="MathJax-Span-693" class="msubsup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.178em, 1000.56em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-694" class="mi">A</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.611em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-695" class="mn">5</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.27em; border-left: 0px solid; width: 0px; height: 1.201em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mn>5</mn></msub></math></span></span><script id="MathJax-Element-156" type="math/tex">A_5</script></span> can be derived from the class multiplication coefficient (see <code class="func">ClassMultiplicationCoefficient</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E2EA9FE7D3062D3"><span class="RefLink">71.12-7</span></a>)) and the number of Galois conjugates of a class (see <code class="func">ClassOrbit</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7ABB007C799F7C49"><span class="RefLink">71.9-12</span></a>)).</p>
-
-<p>The classes <var class="Arg">c1</var>, <var class="Arg">c2</var> and <var class="Arg">c3</var> in the parameter list must be ordered according to the order of the elements in these classes. If elements in class <var class="Arg">c1</var> and <var class="Arg">c2</var> do not generate a polyhedral group then <code class="keyw">fail</code> is returned.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">NrPolyhedralSubgroups( tbl, 2, 2, 4 );</span>
-rec( number := 21, type := "D8" )
-</pre></div>
-
-<p><a id="X7E2EA9FE7D3062D3" name="X7E2EA9FE7D3062D3"></a></p>
-
-<h5>71.12-7 ClassMultiplicationCoefficient</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassMultiplicationCoefficient</code>( <var class="Arg">tbl</var>, <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">k</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>returns the class multiplication coefficient of the classes <var class="Arg">i</var>, <var class="Arg">j</var>, and <var class="Arg">k</var> of the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-157-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-696" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-697" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-698" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-157" type="math/tex">G</script></span> with ordinary character table <var class="Arg">tbl</var>.</p>
-
-<p>The class multiplication coefficient <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-158-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.078em; display: inline-block;" role="math" id="MathJax-Span-699" class="math"><span style="display: inline-block; position: relative; width: 1.587em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.59em, 2.893em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-700" class="mrow"><span id="MathJax-Span-701" class="msubsup"><span style="display: inline-block; position: relative; width: 1.592em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-702" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-703" class="texatom"><span id="MathJax-Span-704" class="mrow"><span id="MathJax-Span-705" class="texatom"><span id="MathJax-Span-706" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-707" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-708" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-709" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-710" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-711" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.448em; border-left: 0px solid; width: 0px; height: 1.084em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></mrow></msub></math></span></span><script id="MathJax-Element-158" type="math/tex">c_{{i,j,k}}</script></span> of the classes <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">k</var> equals the number of pairs <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-159-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.578em; display: inline-block;" role="math" id="MathJax-Span-712" class="math"><span style="display: inline-block; position: relative; width: 1.971em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1001.92em, 2.994em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-713" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-714" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-715" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-716" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-717" class="mi">y</span><span style="font-family: STIXGeneral;" id="MathJax-Span-718" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 1.272em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-159" type="math/tex">(x,y)</script></span> of elements <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi><mo>&#x2208;</mo><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-160-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 4.328em; display: inline-block;" role="math" id="MathJax-Span-719" class="math"><span style="display: inline-block; position: relative; width: 3.317em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1003.32em, 2.994em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-720" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-721" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-722" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-723" class="mi">y</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-724" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-725" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 1.259em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>G</mi></math></span></span><script id="MathJax-Element-160" type="math/tex">x, y \in G</script></span> such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-161-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-726" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.43em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-727" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-728" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.713em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script id="MathJax-Element-161" type="math/tex">x</script></span> lies in class <var class="Arg">i</var>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-162-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.578em; display: inline-block;" role="math" id="MathJax-Span-729" class="math"><span style="display: inline-block; position: relative; width: 0.433em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.41em, 2.994em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-730" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-731" class="mi">y</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 0.966em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span><script id="MathJax-Element-162" type="math/tex">y</script></span> lies in class <var class="Arg">j</var>, and their product <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-163-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.141em; display: inline-block;" role="math" id="MathJax-Span-732" class="math"><span style="display: inline-block; position: relative; width: 0.865em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.85em, 2.994em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-733" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-734" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-735" class="mi">y</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 0.966em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi></math></span></span><script id="MathJax-Element-163" type="math/tex">xy</script></span> is a fixed element of class <var class="Arg">k</var>.</p>
-
-<p>In the center of the group algebra of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-164-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-736" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-737" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-738" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-164" type="math/tex">G</script></span>, these numbers are found as coefficients of the decomposition of the product of two class sums <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>K</mi><mi>i</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-165-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-739" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1000.96em, 2.754em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-740" class="mrow"><span id="MathJax-Span-741" class="msubsup"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-742" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-743" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.179em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>K</mi><mi>i</mi></msub></math></span></span><script id="MathJax-Element-165" type="math/tex">K_i</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>K</mi><mi>j</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-166-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-744" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.655em, 1000.96em, 2.893em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-745" class="mrow"><span id="MathJax-Span-746" class="msubsup"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-747" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-748" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.448em; border-left: 0px solid; width: 0px; height: 1.359em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>K</mi><mi>j</mi></msub></math></span></span><script id="MathJax-Element-166" type="math/tex">K_j</script></span> into class sums:</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"></span><div style="text-align: center;" class="MathJax_Display"><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>K</mi><mi>i</mi></msub><msub><mi>K</mi><mi>j</mi></msub><mo>=</mo><munder><mo>&#x2211;</mo><mi>k</mi></munder><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>K</mi><mi>k</mi></msub><mo>.</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-167-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 9.391em; display: inline-block;" role="math" id="MathJax-Span-749" class="math"><span style="display: inline-block; position: relative; width: 7.212em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.525em, 1007.14em, 4.046em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-750" class="mrow"><span id="MathJax-Span-751" class="msubsup"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-752" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-753" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-754" class="msubsup"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-755" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-756" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-757" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-758" class="munderover"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-759" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.33em, 4.242em, -1000em); top: -2.84em; left: 0.483em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-760" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-761" class="msubsup"><span style="display: inline-block; position: relative; width: 1.239em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-762" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-763" class="texatom"><span id="MathJax-Span-764" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-765" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-766" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-767" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-768" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(3.193em, 1000.72em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-769" class="mi">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.055em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-770" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-771" class="mo">.</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.698em; border-left: 0px solid; width: 0px; height: 3.028em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML MJX_Assistive_MathML_Block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>K</mi><mi>i</mi></msub><msub><mi>K</mi><mi>j</mi></msub><mo>=</mo><munder><mo>∑</mo><mi>k</mi></munder><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>K</mi><mi>k</mi></msub><mo>.</mo></math></span></span></div><script id="MathJax-Element-167" type="math/tex; mode=display">
-K_i K_j = \sum_k c_{ijk} K_k .
-</script></p>
-
-<p>Given the character table of a finite group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-168-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-772" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-773" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-774" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-168" type="math/tex">G</script></span>, whose classes are <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><msub><mi>C</mi><mi>r</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-169-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.453em; display: inline-block;" role="math" id="MathJax-Span-775" class="math"><span style="display: inline-block; position: relative; width: 4.183em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1004.18em, 2.938em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-776" class="mrow"><span id="MathJax-Span-777" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-778" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-779" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-780" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-781" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-782" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-783" class="msubsup"><span style="display: inline-block; position: relative; width: 1.033em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-784" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-785" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.258em; border-left: 0px solid; width: 0px; height: 1.186em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>C</mi><mi>r</mi></msub></math></span></span><script id="MathJax-Element-169" type="math/tex">C_1, \ldots, C_r</script></span> with representatives <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mi>i</mi></msub><mo>&#x2208;</mo><msub><mi>C</mi><mi>i</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-170-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.953em; display: inline-block;" role="math" id="MathJax-Span-786" class="math"><span style="display: inline-block; position: relative; width: 3.029em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1003.03em, 2.994em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-787" class="mrow"><span id="MathJax-Span-788" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-789" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-790" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-791" class="mo">∈</span><span style="padding-left: 0.313em;" id="MathJax-Span-792" class="msubsup"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-793" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-794" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 1.259em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mi>i</mi></msub><mo>∈</mo><msub><mi>C</mi><mi>i</mi></msub></math></span></span><script id="MathJax-Element-170" type="math/tex">g_i \in C_i</script></span>, the class multiplication coefficient <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-171-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.641em; display: inline-block;" role="math" id="MathJax-Span-795" class="math"><span style="display: inline-block; position: relative; width: 1.25em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.867em, 1001.25em, 2.893em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-796" class="mrow"><span id="MathJax-Span-797" class="msubsup"><span style="display: inline-block; position: relative; width: 1.239em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-798" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-799" class="texatom"><span id="MathJax-Span-800" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-801" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-802" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-803" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.448em; border-left: 0px solid; width: 0px; height: 1.084em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></math></span></span><script id="MathJax-Element-171" type="math/tex">c_{ijk}</script></span> can be computed with the following formula:</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"></span><div style="text-align: center;" class="MathJax_Display"><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>C</mi><mi>i</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>&#x22C5;</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>C</mi><mi>j</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>&#x22C5;</mo><munder><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>&#x03C7;</mi><mo>&#x2208;</mo><mi>I</mi><mi>r</mi><mi>r</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></munder><mi>&#x03C7;</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>&#x03C7;</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mi>&#x03C7;</mi><mo stretchy="false">(</mo><msubsup><mi>g</mi><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></mrow></msubsup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>&#x03C7;</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>.</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-172-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 25.703em; display: inline-block;" role="math" id="MathJax-Span-804" class="math"><span style="display: inline-block; position: relative; width: 19.76em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.525em, 1019.69em, 4.185em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-805" class="mrow"><span id="MathJax-Span-806" class="msubsup"><span style="display: inline-block; position: relative; width: 1.239em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.42em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-807" class="mi">c</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-808" class="texatom"><span id="MathJax-Span-809" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-810" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-811" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-812" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-813" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-814" class="texatom"><span id="MathJax-Span-815" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-816" class="mo">|</span></span></span><span id="MathJax-Span-817" class="msubsup"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-818" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-819" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-820" class="texatom"><span id="MathJax-Span-821" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-822" class="mo">|</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-823" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-824" class="texatom"><span id="MathJax-Span-825" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-826" class="mo">|</span></span></span><span id="MathJax-Span-827" class="msubsup"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-828" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-829" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-830" class="texatom"><span id="MathJax-Span-831" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-832" class="mo">|</span></span></span><span id="MathJax-Span-833" class="texatom"><span id="MathJax-Span-834" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-835" class="mo">/</span></span></span><span id="MathJax-Span-836" class="texatom"><span id="MathJax-Span-837" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-838" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-839" class="mi">G</span><span id="MathJax-Span-840" class="texatom"><span id="MathJax-Span-841" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-842" class="mo">|</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-843" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-844" class="munderover"><span style="display: inline-block; position: relative; width: 2.672em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0.69em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-845" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1002.64em, 4.381em, -1000em); top: -2.84em; left: 0em;"><span id="MathJax-Span-846" class="texatom"><span id="MathJax-Span-847" class="mrow"><span id="MathJax-Span-848" class="texatom"><span id="MathJax-Span-849" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-850" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.029em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-851" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-852" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.036em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-853" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-854" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-855" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-856" class="mi">G</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-857" class="mo">)</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-858" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-859" class="mo">(</span><span id="MathJax-Span-860" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-861" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-862" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-863" class="mo">)</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-864" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-865" class="mo">(</span><span id="MathJax-Span-866" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-867" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-868" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-869" class="mo">)</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-870" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-871" class="mo">(</span><span id="MathJax-Span-872" class="msubsup"><span style="display: inline-block; position: relative; width: 1.413em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-873" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.91em, 4.135em, -1000em); top: -4.335em; left: 0.5em;"><span id="MathJax-Span-874" class="texatom"><span id="MathJax-Span-875" class="mrow"><span id="MathJax-Span-876" class="texatom"><span id="MathJax-Span-877" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-878" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-879" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.4em, 4.142em, -1000em); top: -3.672em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-880" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-881" class="mo">)</span><span id="MathJax-Span-882" class="texatom"><span id="MathJax-Span-883" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-884" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-885" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-886" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-887" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-888" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-889" class="mo">.</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.878em; border-left: 0px solid; width: 0px; height: 3.208em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML MJX_Assistive_MathML_Block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>C</mi><mi>i</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>C</mi><mi>j</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>⋅</mo><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>χ</mi><mo>∈</mo><mi>I</mi><mi>r</mi><mi>r</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></munder><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><msubsup><mi>g</mi><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></mrow></msubsup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>χ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>.</mo></math></span></span></div><script id="MathJax-Element-172" type="math/tex; mode=display">
-c_{ijk} = |C_i| \cdot |C_j| / |G| \cdot
-\sum_{{\chi \in Irr(G)}}
-\chi(g_i) \chi(g_j) \chi(g_k^{{-1}}) / \chi(1).
-</script></p>
-
-<p>On the other hand the knowledge of the class multiplication
-coefficients admits the computation of the irreducible characters of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-173-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-890" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-891" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-892" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-173" type="math/tex">G</script></span>, see <code class="func">IrrDixonSchneider</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7ED39DB680BFEA96"><span class="RefLink">71.14-1</span></a>).</p>
-
-<p><a id="X7A19F56C7FD5EFC7" name="X7A19F56C7FD5EFC7"></a></p>
-
-<h5>71.12-8 ClassStructureCharTable</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassStructureCharTable</code>( <var class="Arg">tbl</var>, <var class="Arg">classes</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>returns the so-called class structure of the classes in the list <var class="Arg">classes</var>, for the character table <var class="Arg">tbl</var> of the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-174-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-893" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-894" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-895" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-174" type="math/tex">G</script></span>. The length of <var class="Arg">classes</var> must be at least 2.</p>
-
-<p>Let <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><msub><mi>C</mi><mn>2</mn></msub><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><msub><mi>C</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-175-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 11.016em; display: inline-block;" role="math" id="MathJax-Span-896" class="math"><span style="display: inline-block; position: relative; width: 8.462em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1008.41em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-897" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-898" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-899" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-900" class="mo">(</span><span id="MathJax-Span-901" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-902" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-903" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-904" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-905" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-906" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-907" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-908" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-909" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-910" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-911" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-912" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-913" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-914" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><msub><mi>C</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>C</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-175" type="math/tex">C = (C_1, C_2, \ldots, C_n)</script></span> denote the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-176-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-915" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-916" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-917" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-176" type="math/tex">n</script></span>-tuple of conjugacy classes of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-177-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-918" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-919" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-920" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-177" type="math/tex">G</script></span> that are indexed by <var class="Arg">classes</var>. The class structure <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-178-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.391em; display: inline-block;" role="math" id="MathJax-Span-921" class="math"><span style="display: inline-block; position: relative; width: 1.827em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1001.78em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-922" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-923" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-924" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-925" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-926" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-178" type="math/tex">n(C)</script></span> equals the number of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-179-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-927" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.46em, 2.797em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-928" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-929" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.074em; border-left: 0px solid; width: 0px; height: 0.71em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script id="MathJax-Element-179" type="math/tex">n</script></span>-tuples <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>g</mi><mn>1</mn></msub><mo>,</mo><msub><mi>g</mi><mn>2</mn></msub><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><msub><mi>g</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-180-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.766em; display: inline-block;" role="math" id="MathJax-Span-930" class="math"><span style="display: inline-block; position: relative; width: 5.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1005.91em, 2.994em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-931" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-932" class="mo">(</span><span id="MathJax-Span-933" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-934" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-935" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-936" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-937" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-938" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-939" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-940" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-941" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-942" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-943" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-944" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-945" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-946" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 1.272em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>g</mi><mn>1</mn></msub><mo>,</mo><msub><mi>g</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>g</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-180" type="math/tex">(g_1, g_2, \ldots, g_n)</script></span> of elements <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mi>i</mi></msub><mo>&#x2208;</mo><msub><mi>C</mi><mi>i</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-181-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.953em; display: inline-block;" role="math" id="MathJax-Span-947" class="math"><span style="display: inline-block; position: relative; width: 3.029em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1003.03em, 2.994em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-948" class="mrow"><span id="MathJax-Span-949" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-950" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-951" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-952" class="mo">∈</span><span style="padding-left: 0.313em;" id="MathJax-Span-953" class="msubsup"><span style="display: inline-block; position: relative; width: 0.939em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-954" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-955" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 1.259em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mi>i</mi></msub><mo>∈</mo><msub><mi>C</mi><mi>i</mi></msub></math></span></span><script id="MathJax-Element-181" type="math/tex">g_i \in C_i</script></span> with <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mn>1</mn></msub><msub><mi>g</mi><mn>2</mn></msub><mo>&#x22EF;</mo><msub><mi>g</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-182-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.641em; display: inline-block;" role="math" id="MathJax-Span-956" class="math"><span style="display: inline-block; position: relative; width: 5.865em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1005.76em, 2.994em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-957" class="mrow"><span id="MathJax-Span-958" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-959" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-960" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-961" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-962" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-963" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-964" class="mo">⋯</span><span style="padding-left: 0.188em;" id="MathJax-Span-965" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-966" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-967" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-968" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-969" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 1.272em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mn>1</mn></msub><msub><mi>g</mi><mn>2</mn></msub><mo>⋯</mo><msub><mi>g</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn></math></span></span><script id="MathJax-Element-182" type="math/tex">g_1 g_2 \cdots g_n = 1</script></span>. Note the difference to the definition of the class multiplication coefficients in <code class="func">ClassMultiplicationCoefficient</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E2EA9FE7D3062D3"><span class="RefLink">71.12-7</span></a>).</p>
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-<p><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><msub><mi>C</mi><mn>2</mn></msub><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><msub><mi>C</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-183-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 9.016em; display: inline-block;" role="math" id="MathJax-Span-970" class="math"><span style="display: inline-block; position: relative; width: 6.923em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1006.88em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-971" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-972" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-973" class="mo">(</span><span id="MathJax-Span-974" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-975" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-976" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-977" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-978" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-979" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-980" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-981" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-982" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-983" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-984" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-985" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-986" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-987" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><msub><mi>C</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>C</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-183" type="math/tex">n(C_1, C_2, \ldots, C_n)</script></span> is computed using the formula</p>
-
-<p class="center"><span style="color: inherit;" class="MathJax_Preview"></span><div style="text-align: center;" class="MathJax_Display"><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>n</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><msub><mi>C</mi><mn>2</mn></msub><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><msub><mi>C</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>C</mi><mn>1</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>C</mi><mn>2</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>&#x22EF;</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>C</mi><mi>n</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>&#x22C5;</mo><munder><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>&#x03C7;</mi><mo>&#x2208;</mo><mi>I</mi><mi>r</mi><mi>r</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></munder><mi>&#x03C7;</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>&#x03C7;</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>&#x22EF;</mo><mi>&#x03C7;</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>&#x03C7;</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>2</mn></mrow></mrow></msup><mo>.</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-184-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 39.203em; display: inline-block;" role="math" id="MathJax-Span-988" class="math"><span style="display: inline-block; position: relative; width: 30.144em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.525em, 1030.07em, 4.185em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-989" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-990" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-991" class="mo">(</span><span id="MathJax-Span-992" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-993" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-994" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-995" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-996" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-997" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-998" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-999" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1000" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1001" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1002" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1003" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1004" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1005" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1006" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1007" class="texatom"><span id="MathJax-Span-1008" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1009" class="mo">|</span></span></span><span id="MathJax-Span-1010" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1011" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1012" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1013" class="texatom"><span id="MathJax-Span-1014" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1015" class="mo">|</span></span></span><span id="MathJax-Span-1016" class="texatom"><span id="MathJax-Span-1017" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1018" class="mo">|</span></span></span><span id="MathJax-Span-1019" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1020" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1021" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1022" class="texatom"><span id="MathJax-Span-1023" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1024" class="mo">|</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1025" class="mo">⋯</span><span style="padding-left: 0.188em;" id="MathJax-Span-1026" class="texatom"><span id="MathJax-Span-1027" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1028" class="mo">|</span></span></span><span id="MathJax-Span-1029" class="msubsup"><span style="display: inline-block; position: relative; width: 1.096em; height: 0px;"><span style="position: absolute; clip: rect(3.18em, 1000.69em, 4.153em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1030" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.667em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1031" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1032" class="texatom"><span id="MathJax-Span-1033" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1034" class="mo">|</span></span></span><span id="MathJax-Span-1035" class="texatom"><span id="MathJax-Span-1036" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1037" class="mo">/</span></span></span><span id="MathJax-Span-1038" class="texatom"><span id="MathJax-Span-1039" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1040" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1041" class="mi">G</span><span id="MathJax-Span-1042" class="texatom"><span id="MathJax-Span-1043" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1044" class="mo">|</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1045" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-1046" class="munderover"><span style="display: inline-block; position: relative; width: 2.672em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0.69em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-1047" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1002.64em, 4.381em, -1000em); top: -2.84em; left: 0em;"><span id="MathJax-Span-1048" class="texatom"><span id="MathJax-Span-1049" class="mrow"><span id="MathJax-Span-1050" class="texatom"><span id="MathJax-Span-1051" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1052" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.029em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1053" class="mo">∈</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1054" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.036em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1055" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1056" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.016em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1057" class="mo">(</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1058" class="mi">G</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1059" class="mo">)</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1060" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1061" class="mo">(</span><span id="MathJax-Span-1062" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1063" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1064" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1065" class="mo">)</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1066" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1067" class="mo">(</span><span id="MathJax-Span-1068" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1069" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1070" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1071" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1072" class="mo">⋯</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1073" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1074" class="mo">(</span><span id="MathJax-Span-1075" class="msubsup"><span style="display: inline-block; position: relative; width: 0.929em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.47em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1076" class="mi">g</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1077" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1078" class="mo">)</span><span id="MathJax-Span-1079" class="texatom"><span id="MathJax-Span-1080" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1081" class="mo">/</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1082" class="mi">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.041em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1083" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1084" class="mn">1</span><span id="MathJax-Span-1085" class="msubsup"><span style="display: inline-block; position: relative; width: 1.599em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-1086" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.333em;"><span id="MathJax-Span-1087" class="texatom"><span id="MathJax-Span-1088" class="mrow"><span id="MathJax-Span-1089" class="texatom"><span id="MathJax-Span-1090" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1091" class="mi">n</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1092" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1093" class="mn">2</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1094" class="mo">.</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.878em; border-left: 0px solid; width: 0px; height: 3.208em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML MJX_Assistive_MathML_Block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>n</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><msub><mi>C</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>C</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>C</mi><mn>1</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>C</mi><mn>2</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>⋯</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>C</mi><mi>n</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>⋅</mo><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>χ</mi><mo>∈</mo><mi>I</mi><mi>r</mi><mi>r</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mrow></munder><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>χ</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mrow></msup><mo>.</mo></math></span></span></div><script id="MathJax-Element-184" type="math/tex; mode=display">
-n(C_1, C_2, \ldots, C_n) =
-|C_1| |C_2| \cdots |C_n| / |G| \cdot
-\sum_{{\chi \in Irr(G)}}
-\chi(g_1) \chi(g_2) \cdots \chi(g_n) / \chi(1)^{{n-2}} .
-</script></p>
-
-<p><a id="X809E67E57D4933B3" name="X809E67E57D4933B3"></a></p>
-
-<h5>71.12-9 MatClassMultCoeffsCharTable</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ MatClassMultCoeffsCharTable</code>( <var class="Arg">tbl</var>, <var class="Arg">i</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>For an ordinary character table <var class="Arg">tbl</var> and a class position <var class="Arg">i</var>, <code class="code">MatClassMultCoeffsCharTable</code> returns the matrix <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mo stretchy="false">]</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>,</mo><mi>k</mi></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-185-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.578em; display: inline-block;" role="math" id="MathJax-Span-1095" class="math"><span style="display: inline-block; position: relative; width: 2.74em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.838em, 1002.74em, 3.085em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1096" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1097" class="mo">[</span><span id="MathJax-Span-1098" class="msubsup"><span style="display: inline-block; position: relative; width: 1.296em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.48em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1099" class="mi">a</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.501em;"><span id="MathJax-Span-1100" class="texatom"><span id="MathJax-Span-1101" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1102" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1103" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1104" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-1105" class="msubsup"><span style="display: inline-block; position: relative; width: 1.108em; height: 0px;"><span style="position: absolute; clip: rect(3.184em, 1000.24em, 4.291em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-1106" class="mo">]</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.333em;"><span id="MathJax-Span-1107" class="texatom"><span id="MathJax-Span-1108" class="mrow"><span id="MathJax-Span-1109" class="texatom"><span id="MathJax-Span-1110" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1111" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1112" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1113" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.448em; border-left: 0px solid; width: 0px; height: 1.371em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mo stretchy="false">]</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>,</mo><mi>k</mi></mrow></mrow></msub></math></span></span><script id="MathJax-Element-185" type="math/tex">[ a_{ijk} ]_{{j,k}}</script></span> of structure constants (see <code class="func">ClassMultiplicationCoefficient</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E2EA9FE7D3062D3"><span class="RefLink">71.12-7</span></a>)).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "L3(2)" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassMultiplicationCoefficient( tbl, 2, 2, 4 );</span>
-4
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassStructureCharTable( tbl, [ 2, 2, 4 ] );</span>
-168
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassStructureCharTable( tbl, [ 2, 2, 2, 4 ] );</span>
-1848
-<span class="GAPprompt">gap></span> <span class="GAPinput">MatClassMultCoeffsCharTable( tbl, 2 );</span>
-[ [ 0, 1, 0, 0, 0, 0 ], [ 21, 4, 3, 4, 0, 0 ], [ 0, 8, 6, 8, 7, 7 ],
- [ 0, 8, 6, 1, 7, 7 ], [ 0, 0, 3, 4, 0, 7 ], [ 0, 0, 3, 4, 7, 0 ] ]
-</pre></div>
-
-<p><a id="X7C1941F17BE9FC21" name="X7C1941F17BE9FC21"></a></p>
-
-<h4>71.13 <span class="Heading">Printing Character Tables</span></h4>
-
-<p><a id="X7D45224B86D802E5" name="X7D45224B86D802E5"></a></p>
-
-<h5>71.13-1 ViewObj</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ViewObj</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( method )</td></tr></tbody></table></div>
-<p>The default <code class="func">ViewObj</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) method for ordinary character tables prints the string <code class="code">"CharacterTable"</code>, followed by the identifier (see <code class="func">Identifier</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79C40EE97890202F"><span class="RefLink">71.9-8</span></a>)) or, if known, the group of the character table enclosed in brackets. <code class="func">ViewObj</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) for Brauer tables does the same, except that the first string is replaced by <code class="code">"BrauerTable"</code>, and that the characteristic is also shown.</p>
-
-<p><a id="X836554207C678D41" name="X836554207C678D41"></a></p>
-
-<h5>71.13-2 PrintObj</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ PrintObj</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( method )</td></tr></tbody></table></div>
-<p>The default <code class="func">PrintObj</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) method for character tables does the same as <code class="func">ViewObj</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>), except that <code class="func">PrintObj</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) is used for the group instead of <code class="func">ViewObj</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>).</p>
-
-<p><a id="X7B41F36478C47364" name="X7B41F36478C47364"></a></p>
-
-<h5>71.13-3 Display</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ Display</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( method )</td></tr></tbody></table></div>
-<p>There are various ways to customize the <code class="func">Display</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>) output for character tables. First we describe the default behaviour, alternatives are then described below.</p>
-
-<p>The default <code class="func">Display</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>) method prepares the data in <var class="Arg">tbl</var>
- for a columnwise output. The number of columns printed at one time
-depends on the actual line length, which can be accessed and changed by
-the function <code class="func">SizeScreen</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X8723E0A1837894F3"><span class="RefLink">6.12-1</span></a>).</p>
-
-<p>An interesting variant of <code class="func">Display</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>) is the function <code class="func">PageDisplay</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/doc/chap6_mj.html#X7BB6731F7E3AAA98"><span class="RefLink">GAPDoc: PageDisplay</span></a>). Convenient ways to print the <code class="func">Display</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>) format to a file are given by the function <code class="func">PrintTo1</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/doc/chap6_mj.html#X8603B90C7C3F0AB1"><span class="RefLink">GAPDoc: PrintTo1</span></a>) or by using <code class="func">PageDisplay</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/doc/chap6_mj.html#X7BB6731F7E3AAA98"><span class="RefLink">GAPDoc: PageDisplay</span></a>) and the facilities of the pager used, cf. <code class="func">Pager</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap2_mj.html#X7ED03E41792C3840"><span class="RefLink">2.4-1</span></a>).</p>
-
-<p>An interactive variant of <code class="func">Display</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>) is the <code class="func">Browse</code> (<a href="http://www.gap-system.org/Manuals/pkg/Browse/doc/chap6.html#X7FDD696B7DD54A6E"><span class="RefLink">Browse: Browse</span></a>) method for character tables that is provided by the <strong class="pkg">GAP</strong> package <strong class="pkg">Browse</strong>, see <code class="func">Browse</code> (<a href="http://www.gap-system.org/Manuals/pkg/Browse/doc/chap6.html#X870C744182073CF6"><span class="RefLink">Browse: Browse (for character tables)</span></a>).</p>
-
-<p><code class="func">Display</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>) shows certain characters (by default all irreducible characters) of <var class="Arg">tbl</var>, together with the orders of the centralizers in factorized form and the available power maps (see <code class="func">ComputedPowerMaps</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>)). The <var class="Arg">n</var>-th displayed character is given the name <code class="code">X.<var class="Arg">n</var></code>.</p>
-
-<p>The first lines of the output describe the order of the centralizer
-of an element of the class factorized into its prime divisors.</p>
-
-<p>The next line gives the name of each class. If no class names are stored on <var class="Arg">tbl</var>, <code class="func">ClassNames</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X804CFD597C795801"><span class="RefLink">71.9-6</span></a>) is called.</p>
-
-<p>Preceded by a name <code class="code">P<var class="Arg">n</var></code>, the next lines show the <var class="Arg">n</var>th power maps of <var class="Arg">tbl</var> in terms of the former shown class names.</p>
-
-<p>Every ambiguous or unknown (see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap74_mj.html#X7C1FAB6280A02CCB"><span class="RefLink">74</span></a>) value of the table is displayed as a question mark <code class="code">?</code>.</p>
-
-<p>Irrational character values are not printed explicitly because the
-lengths of their printed representation might disturb the layout.
-Instead of that every irrational value is indicated by a name, which is a
- string of at least one capital letter.</p>
-
-<p>Once a name for an irrational value is found, it is used all over the printed table. Moreover the complex conjugate (see <code class="func">ComplexConjugate</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7BE001A0811CD599"><span class="RefLink">18.5-2</span></a>), <code class="func">GaloisCyc</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X79EE9097783128C4"><span class="RefLink">18.5-1</span></a>)) and the star of an irrationality (see <code class="func">StarCyc</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X7E361C057E97CA66"><span class="RefLink">18.5-3</span></a>)) are represented by that very name preceded by a <code class="code">/</code> and a <code class="code">*</code>, respectively.</p>
-
-<p>The printed character table is then followed by a legend, a list
-identifying the occurring symbols with their actual values. Occasionally
- this identification is supplemented by a quadratic representation of
-the irrationality (see <code class="func">Quadratic</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X84438F867B0CC299"><span class="RefLink">18.5-4</span></a>)) together with the corresponding <strong class="pkg">Atlas</strong> notation (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCCN85">[CCNPW85]</a>).</p>
-
-<p>This default style can be changed by prescribing a record <var class="Arg">arec</var> of options, which can be given</p>
-
-<ol>
-<li><p>as an optional argument in the call to <code class="func">Display</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>),</p>
-
-</li>
-<li><p>as the value of the attribute <code class="func">DisplayOptions</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X85E883A87A190AA4"><span class="RefLink">71.13-4</span></a>) if this value is stored in the table,</p>
-
-</li>
-<li><p>as the value of the global variable <code class="code">CharacterTableDisplayDefaults.User</code>, or</p>
-
-</li>
-<li><p>as the value of the global variable <code class="code">CharacterTableDisplayDefaults.Global</code></p>
-
-</li>
-</ol>
-<p>(in this order of precedence).</p>
-
-<p>The following components of <var class="Arg">arec</var> are supported.</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code">centralizers</code></strong></dt>
-<dd><p><code class="keyw">false</code> to suppress the printing of the orders of the centralizers, or the string <code class="code">"ATLAS"</code> to force the printing of non-factorized centralizer orders in a style similar to that used in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCCN85">[CCNPW85]</a>,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">chars</code></strong></dt>
-<dd><p>an integer or a list of integers to select a sublist of the irreducible characters of <var class="Arg">tbl</var>, or a list of characters of <var class="Arg">tbl</var> (in this case the letter <code class="code">"X"</code> is replaced by <code class="code">"Y"</code>),</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">classes</code></strong></dt>
-<dd><p>an integer or a list of integers to select a sublist of the classes of <var class="Arg">tbl</var>,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">indicator</code></strong></dt>
-<dd><p><code class="keyw">true</code> enables the printing of the second
- Frobenius Schur indicator, a list of integers enables the printing of
-the corresponding indicators (see <code class="func">Indicator</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FD3D3047DE6381E"><span class="RefLink">71.12-5</span></a>)),</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">letter</code></strong></dt>
-<dd><p>a single capital letter (e. g. <code class="code">"P"</code> for permutation characters) to replace the default <code class="code">"X"</code> in character names,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">powermap</code></strong></dt>
-<dd><p>an integer or a list of integers to select a subset of the available power maps, <code class="keyw">false</code> to suppress the printing of power maps, or the string <code class="code">"ATLAS"</code> to force a printing of class names and power maps in a style similar to that used in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCCN85">[CCNPW85]</a>,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">Display</code></strong></dt>
-<dd><p>the function that is actually called in order to display the
-table; the arguments are the table and the optional record, whose
-components can be used inside the <code class="code">Display</code> function,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">StringEntry</code></strong></dt>
-<dd><p>a function that takes either a character value or a character value and the return value of <code class="code">StringEntryData</code>
- (see below), and returns the string that is actually displayed; it is
-called for all character values to be displayed, and also for the
-displayed indicator values (see above),</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">StringEntryData</code></strong></dt>
-<dd><p>a unary function that is called once with argument <var class="Arg">tbl</var> before the character values are displayed; it returns an object that is used as second argument of the function <code class="code">StringEntry</code>,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">Legend</code></strong></dt>
-<dd><p>a function that takes the result of the <code class="code">StringEntryData</code>
- call as its only argument, after the character table has been
-displayed; the return value is a string that describes the symbols used
-in the displayed table in a formatted way, it is printed below the
-displayed table.</p>
-
-</dd>
-</dl>
-<p><a id="X85E883A87A190AA4" name="X85E883A87A190AA4"></a></p>
-
-<h5>71.13-4 DisplayOptions</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ DisplayOptions</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>There is no default method to compute a value, one can set a value with <code class="code">SetDisplayOptions</code>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tbl );</span>
-A5
-
- 2 2 2 . . .
- 3 1 . 1 . .
- 5 1 . . 1 1
-
- 1a 2a 3a 5a 5b
- 2P 1a 1a 3a 5b 5a
- 3P 1a 2a 1a 5b 5a
- 5P 1a 2a 3a 1a 1a
-
-X.1 1 1 1 1 1
-X.2 3 -1 . A *A
-X.3 3 -1 . *A A
-X.4 4 . 1 -1 -1
-X.5 5 1 -1 . .
-
-A = -E(5)-E(5)^4
- = (1-Sqrt(5))/2 = -b5
-<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTableDisplayDefaults.User:= rec(</span>
-<span class="GAPprompt">></span> <span class="GAPinput"> powermap:= "ATLAS", centralizers:= "ATLAS", chars:= false );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Display( CharacterTable( "A5" ) );</span>
-A5
-
- 60 4 3 5 5
-
- p A A A A
- p' A A A A
- 1A 2A 3A 5A B*
-
-<span class="GAPprompt">gap></span> <span class="GAPinput">options:= rec( chars:= 4, classes:= [ tbl.3a .. tbl.5a ],</span>
-<span class="GAPprompt">></span> <span class="GAPinput"> centralizers:= false, indicator:= true,</span>
-<span class="GAPprompt">></span> <span class="GAPinput"> powermap:= [ 2 ] );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tbl, options );</span>
-A5
-
- 3a 5a
- 2P 3a 5b
- 2
-X.4 + 1 -1
-<span class="GAPprompt">gap></span> <span class="GAPinput">SetDisplayOptions( tbl, options ); Display( tbl );</span>
-A5
-
- 3a 5a
- 2P 3a 5b
- 2
-X.4 + 1 -1
-<span class="GAPprompt">gap></span> <span class="GAPinput">Unbind( CharacterTableDisplayDefaults.User );</span>
-</pre></div>
-
-<p><a id="X79EC9603833AA2AB" name="X79EC9603833AA2AB"></a></p>
-
-<h5>71.13-5 PrintCharacterTable</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ PrintCharacterTable</code>( <var class="Arg">tbl</var>, <var class="Arg">varname</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">tbl</var> be a nearly character table, and <var class="Arg">varname</var> a string. <code class="func">PrintCharacterTable</code> prints those values of the supported attributes (see <code class="func">SupportedCharacterTableInfo</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7DBEF4BF87F10CD6"><span class="RefLink">71.3-4</span></a>)) that are known for <var class="Arg">tbl</var>.</p>
-
-<p>The output of <code class="func">PrintCharacterTable</code> is <strong class="pkg">GAP</strong> readable; actually reading it into <strong class="pkg">GAP</strong> will bind the variable with name <var class="Arg">varname</var> to a character table that coincides with <var class="Arg">tbl</var> for all printed components.</p>
-
-<p>This is used mainly for saving character tables to files. A more human readable form is produced by <code class="func">Display</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></a>).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">PrintCharacterTable( CharacterTable( "Cyclic", 2 ), "tbl" );</span>
-tbl:= function()
-local tbl, i;
-tbl:=rec();
-tbl.Irr:=
-[ [ 1, 1 ], [ 1, -1 ] ];
-tbl.NrConjugacyClasses:=
-2;
-tbl.Size:=
-2;
-tbl.OrdersClassRepresentatives:=
-[ 1, 2 ];
-tbl.SizesCentralizers:=
-[ 2, 2 ];
-tbl.UnderlyingCharacteristic:=
-0;
-tbl.ClassParameters:=
-[ [ 1, 0 ], [ 1, 1 ] ];
-tbl.CharacterParameters:=
-[ [ 1, 0 ], [ 1, 1 ] ];
-tbl.Identifier:=
-"C2";
-tbl.InfoText:=
-"computed using generic character table for cyclic groups";
-tbl.ComputedPowerMaps:=
-[ , [ 1, 1 ] ];
-ConvertToLibraryCharacterTableNC(tbl);
-return tbl;
-end;
-tbl:= tbl();
-</pre></div>
-
-<p><a id="X79BC08C6846718D9" name="X79BC08C6846718D9"></a></p>
-
-<h4>71.14 <span class="Heading">Computing the Irreducible Characters of a Group</span></h4>
-
-<p>Several algorithms are available for computing the irreducible characters of a finite group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-186-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1114" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1115" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1116" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-186" type="math/tex">G</script></span>. The default method for arbitrary finite groups is to use the Dixon-Schneider algorithm (see <code class="func">IrrDixonSchneider</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7ED39DB680BFEA96"><span class="RefLink">71.14-1</span></a>)). For supersolvable groups, Conlon's algorithm can be used (see <code class="func">IrrConlon</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E81BCD686561DF0"><span class="RefLink">71.14-2</span></a>)). For abelian-by-supersolvable groups, the Baum-Clausen algorithm for computing the irreducible representations (see <code class="func">IrreducibleRepresentations</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7F29C5447B5DC102"><span class="RefLink">71.14-4</span></a>)) can be used to compute the irreducible characters (see <code class="func">IrrBaumClausen</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7BF15729839203FC"><span class="RefLink">71.14-3</span></a>)).</p>
-
-<p>These functions are installed in methods for <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>), but explicitly calling one of them will <em>not</em> set the <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) value of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-187-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1117" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1118" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1119" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-187" type="math/tex">G</script></span>.</p>
-
-<p><a id="X7ED39DB680BFEA96" name="X7ED39DB680BFEA96"></a></p>
-
-<h5>71.14-1 IrrDixonSchneider</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IrrDixonSchneider</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>computes the irreducible characters of the finite group <var class="Arg">G</var>, using the Dixon-Schneider method (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86CDA4007A5EF704"><span class="RefLink">71.16</span></a>). It calls <code class="func">DixonInit</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E33C03E7BDDC9B0"><span class="RefLink">71.17-2</span></a>) and <code class="func">DixonSplit</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87ABE0B081DAD476"><span class="RefLink">71.17-4</span></a>), and finally returns the list returned by <code class="func">DixontinI</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X868476037907918F"><span class="RefLink">71.17-3</span></a>). See also the sections <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C1153637E7D2133"><span class="RefLink">71.18</span></a> and <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X782B5E37848786BC"><span class="RefLink">71.19</span></a>.</p>
-
-<p><a id="X7E81BCD686561DF0" name="X7E81BCD686561DF0"></a></p>
-
-<h5>71.14-2 IrrConlon</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IrrConlon</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For a finite solvable group <var class="Arg">G</var>, <code class="func">IrrConlon</code> returns a list of certain irreducible characters of <var class="Arg">G</var>, among those all irreducibles that have the supersolvable residuum of <var class="Arg">G</var> in their kernels; so if <var class="Arg">G</var> is supersolvable, all irreducible characters of <var class="Arg">G</var> are returned. An error is signalled if <var class="Arg">G</var> is not solvable.</p>
-
-<p>The characters are computed using Conlon's algorithm (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCon90a">[Con90a]</a> and <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCon90b">[Con90b]</a>). For each irreducible character in the returned list, the monomiality information (see <code class="func">TestMonomial</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap75_mj.html#X84EB92B57DAF5C93"><span class="RefLink">75.4-1</span></a>)) is stored.</p>
-
-<p><a id="X7BF15729839203FC" name="X7BF15729839203FC"></a></p>
-
-<h5>71.14-3 IrrBaumClausen</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IrrBaumClausen</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p><code class="func">IrrBaumClausen</code> returns the absolutely irreducible ordinary characters of the factor group of the finite solvable group <var class="Arg">G</var> by the derived subgroup of its supersolvable residuum.</p>
-
-<p>The characters are computed using the algorithm by Baum and Clausen (see <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBBC94">[BC94]</a>). An error is signalled if <var class="Arg">G</var> is not solvable.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">g:= SL(2,3);;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">irr1:= IrrDixonSchneider( g );</span>
-[ Character( CharacterTable( SL(2,3) ), [ 1, 1, 1, 1, 1, 1, 1 ] ),
- Character( CharacterTable( SL(2,3) ),
- [ 1, E(3)^2, E(3), 1, E(3), E(3)^2, 1 ] ),
- Character( CharacterTable( SL(2,3) ),
- [ 1, E(3), E(3)^2, 1, E(3)^2, E(3), 1 ] ),
- Character( CharacterTable( SL(2,3) ), [ 2, 1, 1, -2, -1, -1, 0 ] ),
- Character( CharacterTable( SL(2,3) ),
- [ 2, E(3)^2, E(3), -2, -E(3), -E(3)^2, 0 ] ),
- Character( CharacterTable( SL(2,3) ),
- [ 2, E(3), E(3)^2, -2, -E(3)^2, -E(3), 0 ] ),
- Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">irr2:= IrrConlon( g );</span>
-[ Character( CharacterTable( SL(2,3) ), [ 1, 1, 1, 1, 1, 1, 1 ] ),
- Character( CharacterTable( SL(2,3) ),
- [ 1, E(3), E(3)^2, 1, E(3)^2, E(3), 1 ] ),
- Character( CharacterTable( SL(2,3) ),
- [ 1, E(3)^2, E(3), 1, E(3), E(3)^2, 1 ] ),
- Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">irr3:= IrrBaumClausen( g );</span>
-[ Character( CharacterTable( SL(2,3) ), [ 1, 1, 1, 1, 1, 1, 1 ] ),
- Character( CharacterTable( SL(2,3) ),
- [ 1, E(3), E(3)^2, 1, E(3)^2, E(3), 1 ] ),
- Character( CharacterTable( SL(2,3) ),
- [ 1, E(3)^2, E(3), 1, E(3), E(3)^2, 1 ] ),
- Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">chi:= irr2[4];; HasTestMonomial( chi );</span>
-true
-</pre></div>
-
-<p><a id="X7F29C5447B5DC102" name="X7F29C5447B5DC102"></a></p>
-
-<h5>71.14-4 IrreducibleRepresentations</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IrreducibleRepresentations</code>( <var class="Arg">G</var>[, <var class="Arg">F</var>] )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>Called with a finite group <var class="Arg">G</var> and a field <var class="Arg">F</var>, <code class="func">IrreducibleRepresentations</code> returns a list of representatives of the irreducible matrix representations of <var class="Arg">G</var> over <var class="Arg">F</var>, up to equivalence.</p>
-
-<p>If <var class="Arg">G</var> is the only argument then <code class="func">IrreducibleRepresentations</code> returns a list of representatives of the absolutely irreducible complex representations of <var class="Arg">G</var>, up to equivalence.</p>
-
-<p>At the moment, methods are available for the following cases: If <var class="Arg">G</var> is abelian by supersolvable the method of <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBBC94">[BC94]</a> is used.</p>
-
-<p>Otherwise, if <var class="Arg">F</var> and <var class="Arg">G</var> are both finite, the regular module of <var class="Arg">G</var> is split by MeatAxe methods which can make this an expensive operation.</p>
-
-<p>Finally, if <var class="Arg">F</var> is not given (i.e. it defaults to the cyclotomic numbers) and <var class="Arg">G</var> is a finite group, the method of <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBDix93">[Dix93]</a> (see <code class="func">IrreducibleRepresentationsDixon</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8493ED7A86FFCB8A"><span class="RefLink">71.14-5</span></a>)) is used.</p>
-
-<p>For other cases no methods are implemented yet.</p>
-
-<p>The representations obtained are <em>not</em> guaranteed to be "nice" (for example preserving a unitary form) in any way.</p>
-
-<p>See also <code class="func">IrreducibleModules</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87E82F8085745523"><span class="RefLink">71.15-1</span></a>), which provides efficient methods for solvable groups.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">g:= AlternatingGroup( 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= IrreducibleRepresentations( g );</span>
-[ Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
- [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
- Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
- [ [ [ E(3) ] ], [ [ 1 ] ], [ [ 1 ] ] ],
- Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
- [ [ [ E(3)^2 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
- Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
- [ [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
- [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ],
- [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( repr, IsGroupHomomorphism );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">Length( repr );</span>
-4
-<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( g );</span>
-[ (1,2,3), (2,3,4) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( gens, x -> x^repr[1] );</span>
-[ [ [ 1 ] ], [ [ 1 ] ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput"> List( gens, x -> x^repr[4] );</span>
-[ [ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ],
- [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]
-</pre></div>
-
-<p><a id="X8493ED7A86FFCB8A" name="X8493ED7A86FFCB8A"></a></p>
-
-<h5>71.14-5 IrreducibleRepresentationsDixon</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IrreducibleRepresentationsDixon</code>( <var class="Arg">G</var>[, <var class="Arg">chi</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>Called with one argument, a group <var class="Arg">G</var>, <code class="func">IrreducibleRepresentationsDixon</code> computes (representatives of) all irreducible complex representations for the finite group <var class="Arg">G</var>, using the method of <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBDix93">[Dix93]</a>,
- which computes the character table and computes the representation as
-constituent of an induced monomial representation of a subgroup.</p>
-
-<p>This method can be quite expensive for larger groups, for example it might involve calculation of the subgroup lattice of <var class="Arg">G</var>.</p>
-
-<p>A character <var class="Arg">chi</var> of <var class="Arg">G</var> can be given as the second argument, in this case only a representation affording <var class="Arg">chi</var> is returned.</p>
-
-<p>The second argument can also be a list of characters of <var class="Arg">G</var>, in this case only representations for characters in this list are computed.</p>
-
-<p>Note that this method might fail if for an irreducible representation
- there is no subgroup in which its reduction has a linear constituent
-with multiplicity one.</p>
-
-<p>If the option <var class="Arg">unitary</var> is given, <strong class="pkg">GAP</strong> tries, at extra cost, to find a unitary representation (and will issue an error if it cannot do so).</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">a5:= AlternatingGroup( 5 );</span>
-Alt( [ 1 .. 5 ] )
-<span class="GAPprompt">gap></span> <span class="GAPinput">char:= First( Irr( a5 ), x -> x[1] = 4 );</span>
-Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 4, 0, 1, -1, -1 ] )
-<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=IrreducibleRepresentationsDixon( a5, char: unitary );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Order( a5.1*a5.2 ) = Order( Image( hom, a5.1 )*Image( hom, a5.2 ) );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">reps:= List( ConjugacyClasses( a5 ), Representative );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( reps, g -> TraceMat( Image( hom, g ) ) );</span>
-[ 4, 0, 1, -1, -1 ]
-</pre></div>
-
-<p><a id="X7E51AACD79CE0BC8" name="X7E51AACD79CE0BC8"></a></p>
-
-<h4>71.15 <span class="Heading">Representations Given by Modules</span></h4>
-
-<p>This section describes functions that return certain modules of a
-given group. (Extensions by modules can be formed by the command <code class="func">Extensions</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap46_mj.html#X8236AD927A5A0E5A"><span class="RefLink">46.8-4</span></a>).)</p>
-
-<p><a id="X87E82F8085745523" name="X87E82F8085745523"></a></p>
-
-<h5>71.15-1 IrreducibleModules</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IrreducibleModules</code>( <var class="Arg">G</var>, <var class="Arg">F</var>, <var class="Arg">dim</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>returns a list of length 2. The first entry is a generating system of <var class="Arg">G</var>. The second entry is a list of all irreducible modules of <var class="Arg">G</var> over the field <var class="Arg">F</var> in dimension <var class="Arg">dim</var>, given as MeatAxe modules (see <code class="func">GModuleByMats</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap69_mj.html#X801022027B066497"><span class="RefLink">69.1-1</span></a>)).</p>
-
-<p><a id="X7D0BD5337D1C069B" name="X7D0BD5337D1C069B"></a></p>
-
-<h5>71.15-2 AbsolutelyIrreducibleModules</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ AbsolutelyIrreducibleModules</code>( <var class="Arg">G</var>, <var class="Arg">F</var>, <var class="Arg">dim</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ AbsoluteIrreducibleModules</code>( <var class="Arg">G</var>, <var class="Arg">F</var>, <var class="Arg">dim</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ AbsolutIrreducibleModules</code>( <var class="Arg">G</var>, <var class="Arg">F</var>, <var class="Arg">dim</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>returns a list of length 2. The first entry is a generating system of <var class="Arg">G</var>. The second entry is a list of all absolute irreducible modules of <var class="Arg">G</var> over the field <var class="Arg">F</var> in dimension <var class="Arg">dim</var>, given as MeatAxe modules (see <code class="func">GModuleByMats</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap69_mj.html#X801022027B066497"><span class="RefLink">69.1-1</span></a>)). The other two names are just synonyms.</p>
-
-<p><a id="X7EB88B2E87AF5556" name="X7EB88B2E87AF5556"></a></p>
-
-<h5>71.15-3 RegularModule</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ RegularModule</code>( <var class="Arg">G</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>returns a list of length 2. The first entry is a generating system of <var class="Arg">G</var>. The second entry is the regular module of <var class="Arg">G</var> over <var class="Arg">F</var>, given as a MeatAxe module (see <code class="func">GModuleByMats</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap69_mj.html#X801022027B066497"><span class="RefLink">69.1-1</span></a>)).</p>
-
-<p><a id="X86CDA4007A5EF704" name="X86CDA4007A5EF704"></a></p>
-
-<h4>71.16 <span class="Heading">The Dixon-Schneider Algorithm</span></h4>
-
-<p>The <strong class="pkg">GAP</strong> library implementation of the
-Dixon-Schneider algorithm first computes the linear characters, using
-the commutator factor group. If irreducible characters are missing
-afterwards, they are computed using the techniques described in <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBDix67">[Dix67]</a>, <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBSch90">[Sch90]</a> and <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBHulpke93">[Hul93]</a>.</p>
-
-<p>Called with a group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-188-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1120" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1121" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1122" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-188" type="math/tex">G</script></span>, the function <code class="func">CharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FCA7A7A822BDA33"><span class="RefLink">71.3-1</span></a>)
- returns a character table object that stores already information such
-as class lengths, but not the irreducible characters. The routines that
-compute the irreducibles may use the information that is already
-contained in this table object. In particular the ordering of classes in
- the computed characters coincides with the ordering of classes in the
-character table of <var class="Arg">G</var> (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X793E0EBF84B07313"><span class="RefLink">71.6</span></a>). Thus it is possible to combine computations using the group with character theoretic computations (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C083207868066C1"><span class="RefLink">71.17</span></a>
- for details), for example one can enter known characters. Note that the
- user is responsible for the correctness of the characters. (There is
-little use in providing the trivial character to the routine.)</p>
-
-<p>The computation of irreducible characters from the group needs to
-identify the classes of group elements very often, so it can be helpful
-to store a class list of all group elements. Since this is obviously
-limited by the group order, it is controlled by the global function <code class="func">IsDxLargeGroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8089009E7EA85BC8"><span class="RefLink">71.17-8</span></a>).</p>
-
-<p>The routines compute in a prime field of size <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-189-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1123" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.059em, 1000.45em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1124" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1125" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 0.965em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script id="MathJax-Element-189" type="math/tex">p</script></span>, such that the exponent of the group divides <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>p</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-190-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.641em; display: inline-block;" role="math" id="MathJax-Span-1126" class="math"><span style="display: inline-block; position: relative; width: 2.788em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.74em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1127" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1128" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1129" class="mi">p</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1130" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1131" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1132" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.27em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-190" type="math/tex">(p-1)</script></span> and such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></msqrt><mo>&lt;</mo><mi>p</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-191-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 5.391em; display: inline-block;" role="math" id="MathJax-Span-1133" class="math"><span style="display: inline-block; position: relative; width: 4.135em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.679em, 1004.1em, 2.993em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1134" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1135" class="mn">2</span><span id="MathJax-Span-1136" class="msqrt"><span style="display: inline-block; position: relative; width: 1.884em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1001.05em, 4.153em, -1000em); top: -3.99em; left: 0.737em;"><span id="MathJax-Span-1137" class="mrow"><span id="MathJax-Span-1138" class="texatom"><span id="MathJax-Span-1139" class="mrow"><span id="MathJax-Span-1140" class="texatom"><span id="MathJax-Span-1141" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1142" class="mo">|</span></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1143" class="mi">G</span><span id="MathJax-Span-1144" class="texatom"><span id="MathJax-Span-1145" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1146" class="mo">|</span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.026em, 1001.15em, 3.365em, -1000em); top: -3.991em; left: 0.737em;"><span style="display: inline-block; position: relative; width: 1.147em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.99em; left: 0.647em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.99em; left: 0.311em;">‾<span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(2.903em, 1000.77em, 4.146em, -1000em); top: -3.868em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1147" class="mo"><</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1148" class="mi">p</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.329em; border-left: 0px solid; width: 0px; height: 1.459em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></msqrt><mo><</mo><mi>p</mi></math></span></span><script id="MathJax-Element-191" type="math/tex">2 \sqrt{{|G|}} < p</script></span>. Currently prime fields of size smaller than <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>65</mn><mspace width="thinmathspace" /><mn>536</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-192-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 3.391em; display: inline-block;" role="math" id="MathJax-Span-1149" class="math"><span style="display: inline-block; position: relative; width: 2.596em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.812em, 1002.56em, 2.802em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1150" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1151" class="mn">65</span><span style="height: 0em; vertical-align: 0em; width: 0.188em; display: inline-block; overflow: hidden;" id="MathJax-Span-1152" class="mspace"></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1153" class="mn">536</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.081em; border-left: 0px solid; width: 0px; height: 1.038em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>65</mn><mspace width="thinmathspace"></mspace><mn>536</mn></math></span></span><script id="MathJax-Element-192" type="math/tex">65\,536</script></span>
- are handled more efficiently than larger prime fields, so the runtime
-of the character calculation depends on how large the chosen prime is.</p>
-
-<p>The routine stores a Dixon record (see <code class="func">DixonRecord</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C398F2680C8616B"><span class="RefLink">71.17-1</span></a>)) in the group that helps routines that identify classes, for example <code class="func">FusionConjugacyClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X86CE53B681F13C63"><span class="RefLink">73.3-1</span></a>), to work much faster. Note that interrupting Dixon-Schneider calculations will prevent <strong class="pkg">GAP</strong> from cleaning up the Dixon record; when the computation by <code class="func">IrrDixonSchneider</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7ED39DB680BFEA96"><span class="RefLink">71.14-1</span></a>) is complete, the possibly large record is shrunk to an acceptable size.</p>
-
-<p><a id="X7C083207868066C1" name="X7C083207868066C1"></a></p>
-
-<h4>71.17 <span class="Heading">Advanced Methods for Dixon-Schneider Calculations</span></h4>
-
-<p>The computation of irreducible characters of very large groups may
-take quite some time. On the other hand, for the expert only a few
-irreducible characters may be needed, since the other ones can be
-computed using character theoretic methods such as tensoring, induction,
- and restriction. Thus <strong class="pkg">GAP</strong> provides also
-step-by-step routines for doing the calculations. These routines allow
-one to compute some characters and to stop before all are calculated.
-Note that there is no "safety net": The routines (being somehow
-internal) do no error checking, and assume the information given is
-correct.</p>
-
-<p>When the info level of <code class="func">InfoCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C6F3D947E5188D1"><span class="RefLink">71.4-2</span></a>) if positive, information about the progress of splitting is printed. (The default value is zero.)</p>
-
-<p><a id="X7C398F2680C8616B" name="X7C398F2680C8616B"></a></p>
-
-<h5>71.17-1 DixonRecord</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ DixonRecord</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>The <code class="func">DixonRecord</code> of a group contains
-information used by the routines to compute the irreducible characters
-and related information via the Dixon-Schneider algorithm such as class
-arrangement and character spaces split obtained so far. Usually this
-record is passed as argument to all subfunctions to avoid a long
-argument list. It has a component <code class="code">conjugacyClasses</code> which contains the classes of <var class="Arg">G</var> <em>ordered as the algorithm needs them</em>.</p>
-
-<p><a id="X7E33C03E7BDDC9B0" name="X7E33C03E7BDDC9B0"></a></p>
-
-<h5>71.17-2 DixonInit</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ DixonInit</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>This function does all the initializations for the Dixon-Schneider
-algorithm. This includes calculation of conjugacy classes, power maps,
-linear characters and character morphisms. It returns a record
-(see <code class="func">DixonRecord</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C398F2680C8616B"><span class="RefLink">71.17-1</span></a>) and Section <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C1153637E7D2133"><span class="RefLink">71.18</span></a>) that can be used when calculating the irreducible characters of <var class="Arg">G</var> interactively.</p>
-
-<p><a id="X868476037907918F" name="X868476037907918F"></a></p>
-
-<h5>71.17-3 DixontinI</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ DixontinI</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>This function ends a Dixon-Schneider calculation. It sorts the
-characters according to the degree and unbinds components in the Dixon
-record that are not of use any longer. It returns a list of irreducible
-characters.</p>
-
-<p><a id="X87ABE0B081DAD476" name="X87ABE0B081DAD476"></a></p>
-
-<h5>71.17-4 DixonSplit</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ DixonSplit</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>This function performs one splitting step in the Dixon-Schneider
-algorithm. It selects a class, computes the (partial) class sum matrix,
-uses it to split character spaces and stores all the irreducible
-characters obtained that way.</p>
-
-<p>The class to use for splitting is chosen via <code class="func">BestSplittingMatrix</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7BFD4C1A821731FB"><span class="RefLink">71.17-5</span></a>) and the options described for this function apply here.</p>
-
-<p><code class="func">DixonSplit</code> returns the number of the class that was used for splitting if a split was performed, and <code class="keyw">fail</code> otherwise.</p>
-
-<p><a id="X7BFD4C1A821731FB" name="X7BFD4C1A821731FB"></a></p>
-
-<h5>71.17-5 BestSplittingMatrix</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ BestSplittingMatrix</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>returns the number of the class sum matrix that is assumed to yield
-the best (cost/earning ration) split. This matrix then will be the next
-one computed and used.</p>
-
-<p>The global option <code class="code">maxclasslen</code> (defaulting to <code class="func">infinity</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>)) is recognized by <code class="func">BestSplittingMatrix</code>:
- Only classes whose length is limited by the value of this option will
-be considered for splitting. If no usable class remains, <code class="keyw">fail</code> is returned.</p>
-
-<p><a id="X7C85B56C80BFA2E3" name="X7C85B56C80BFA2E3"></a></p>
-
-<h5>71.17-6 DxIncludeIrreducibles</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ DxIncludeIrreducibles</code>( <var class="Arg">D</var>, <var class="Arg">new</var>[, <var class="Arg">newmod</var>] )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>This function takes a list of irreducible characters <var class="Arg">new</var>, each given as a list of values (corresponding to the class arrangement in <var class="Arg">D</var>), and adds these to a partial computed list of irreducibles as maintained by the Dixon record <var class="Arg">D</var>.
- This permits one to add characters in interactive use obtained from
-other sources and to continue the Dixon-Schneider calculation
-afterwards. If the optional argument <var class="Arg">newmod</var> is given, it must be a list of reduced characters, corresponding to <var class="Arg">new</var>. (Otherwise the function has to reduce the characters itself.)</p>
-
-<p>The function closes the new characters under the action of Galois automorphisms and tensor products with linear characters.</p>
-
-<p><a id="X87A5B5C77F7F348E" name="X87A5B5C77F7F348E"></a></p>
-
-<h5>71.17-7 SplitCharacters</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SplitCharacters</code>( <var class="Arg">D</var>, <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>This routine decomposes the characters given in <var class="Arg">list</var>
- according to the character spaces found up to this point. By applying
-this routine to tensor products etc., it may result in characters with
-smaller norm, even irreducible ones. Since the recalculation of
-characters is only possible if the degree is small enough, the splitting
- process is applied only to characters of sufficiently small degree.</p>
-
-<p><a id="X8089009E7EA85BC8" name="X8089009E7EA85BC8"></a></p>
-
-<h5>71.17-8 IsDxLargeGroup</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ IsDxLargeGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>returns <code class="keyw">true</code> if the order of the group <var class="Arg">G</var> is smaller than the current value of the global variable <code class="code">DXLARGEGROUPORDER</code>, and <code class="keyw">false</code>
- otherwise. In Dixon-Schneider calculations, for small groups in the
-above sense a class map is stored, whereas for large groups, each
-occurring element is identified individually.</p>
-
-<p><a id="X7C1153637E7D2133" name="X7C1153637E7D2133"></a></p>
-
-<h4>71.18 <span class="Heading">Components of a Dixon Record</span></h4>
-
-<p>The "Dixon record" <var class="Arg">D</var> returned by <code class="func">DixonInit</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E33C03E7BDDC9B0"><span class="RefLink">71.17-2</span></a>)
- stores all the information that is used by the Dixon-Schneider routines
- while computing the irreducible characters of a group. Some entries,
-however, may be useful to know about when using the algorithm
-interactively, see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X782B5E37848786BC"><span class="RefLink">71.19</span></a>.</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code">group</code></strong></dt>
-<dd><p>the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-193-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1154" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1155" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1156" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-193" type="math/tex">G</script></span> of which the character table is to be computed,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">conjugacyClasses</code></strong></dt>
-<dd><p>classes of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-194-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1157" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1158" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1159" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-194" type="math/tex">G</script></span> (all characters stored in the Dixon record correspond to this arrangement of classes),</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">irreducibles</code></strong></dt>
-<dd><p>the already known irreducible characters (given as lists of their values on the conjugacy classes),</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">characterTable</code></strong></dt>
-<dd><p>the <code class="func">CharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7FCA7A7A822BDA33"><span class="RefLink">71.3-1</span></a>) value of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-195-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1160" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1161" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1162" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-195" type="math/tex">G</script></span> (whose irreducible characters are not yet known),</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">ClassElement( <var class="Arg">D</var>, <var class="Arg">el</var> )</code></strong></dt>
-<dd><p>a function that returns the number of the class of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-196-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1163" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1164" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1165" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-196" type="math/tex">G</script></span> that contains the element <var class="Arg">el</var>.</p>
-
-</dd>
-</dl>
-<p><a id="X782B5E37848786BC" name="X782B5E37848786BC"></a></p>
-
-<h4>71.19 <span class="Heading">An Example of Advanced Dixon-Schneider Calculations</span></h4>
-
-<p>First, we set the appropriate info level higher.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel( InfoCharacterTable, 1 );</span>
-</pre></div>
-
-<p>for printout of some internal results. We now define our group, which is isomorphic to PSL<span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi></mi><mn>4</mn></msub><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-197-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.078em; display: inline-block;" role="math" id="MathJax-Span-1166" class="math"><span style="display: inline-block; position: relative; width: 1.587em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1001.54em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1167" class="mrow"><span id="MathJax-Span-1168" class="msubsup"><span style="display: inline-block; position: relative; width: 0.429em; height: 0px;"><span style="position: absolute; clip: rect(3.846em, 1000em, 4.135em, -1000em); top: -3.99em; left: 0em;"><span id="MathJax-Span-1169" class="mi"></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1170" class="mn">4</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1171" class="mo">(</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1172" class="mn">3</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1173" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.234em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi></mi><mn>4</mn></msub><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math></span></span><script id="MathJax-Element-197" type="math/tex">_4(3)</script></span>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">g:= PrimitiveGroup(40,5);</span>
-PSL(4, 3)
-<span class="GAPprompt">gap></span> <span class="GAPinput">Size(g);</span>
-6065280
-<span class="GAPprompt">gap></span> <span class="GAPinput">d:= DixonInit( g );;</span>
-#I 29 classes
-#I choosing prime 65521
-<span class="GAPprompt">gap></span> <span class="GAPinput">c:= d.characterTable;;</span>
-</pre></div>
-
-<p>After the initialisation, one structure matrix is evaluated, yielding smaller spaces and several irreducible characters.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">DixonSplit( d );</span>
-#I Matrix 2,Representative of Order 3,Centralizer: 5832
-#I Dimensions: [ 1, 2, 1, 4, 12, 1, 1, 2, 1, 2, 1 ]
-#I Two-dim space split
-#I Two-dim space split
-#I Two-dim space split
-2
-</pre></div>
-
-<p>In this case spaces of the listed dimensions are a result of the
-splitting process. The three two dimensional spaces are split
-successfully by combinatoric means.</p>
-
-<p>We obtain several irreducible characters by tensor products and notify them to the Dixon record.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">asp:= AntiSymmetricParts( c, d.irreducibles, 2 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ro:= ReducedCharacters( c, d.irreducibles, asp );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Length( ro.irreducibles );</span>
-3
-<span class="GAPprompt">gap></span> <span class="GAPinput">DxIncludeIrreducibles( d, ro.irreducibles );</span>
-</pre></div>
-
-<p>The tensor products of the nonlinear characters among each other are
-reduced with the irreducible characters. The result is split according
-to the spaces found, which yields characters of smaller norms, but no
-new irreducibles.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">nlc:= Filtered( d.irreducibles, i -> i[1] > 1 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">t:= Tensored( nlc, nlc );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ro:= ReducedCharacters( c, d.irreducibles, t );; ro.irreducibles;</span>
-[ ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( ro.remainders, i -> ScalarProduct( c, i, i) );</span>
-[ 2, 2, 4, 4, 4, 4, 13, 13, 18, 18, 19, 21, 21, 36, 36, 29, 34, 34,
- 42, 34, 48, 54, 62, 68, 68, 78, 84, 84, 88, 90, 159, 169, 169, 172,
- 172, 266, 271, 271, 268, 274, 274, 280, 328, 373, 373, 456, 532,
- 576, 679, 683, 683, 754, 768, 768, 890, 912, 962, 1453, 1453, 1601,
- 1601, 1728, 1739, 1739, 1802, 2058, 2379, 2414, 2543, 2744, 2744,
- 2920, 3078, 3078, 4275, 4275, 4494, 4760, 5112, 5115, 5115, 5414,
- 6080, 6318, 7100, 7369, 7369, 7798, 8644, 10392, 12373, 12922,
- 14122, 14122, 18948, 21886, 24641, 24641, 25056, 38942, 44950,
- 78778 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">t:= SplitCharacters( d, ro.remainders );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( t, i -> ScalarProduct( c, i, i ) );</span>
-[ 2, 2, 4, 2, 2, 4, 4, 3, 6, 5, 5, 9, 9, 4, 12, 13, 18, 18, 20, 18,
- 20, 24, 26, 32, 32, 16, 42, 59, 69, 69, 72, 72, 36, 72, 78, 78, 84,
- 122, 117, 127, 117, 127, 64, 132, 100, 144, 196, 256, 456, 532,
- 576, 679, 683, 683, 754, 768, 768, 890, 912, 962, 1453, 1453, 1601,
- 1601, 1728, 1739, 1739, 1802, 2058, 2379, 2414, 2543, 2744, 2744,
- 2920, 3078, 3078, 4275, 4275, 4494, 4760, 5112, 5115, 5115, 5414,
- 6080, 6318, 7100, 7369, 7369, 7798, 8644, 10392, 12373, 12922,
- 14122, 14122, 18948, 21886, 24641, 24641, 25056, 38942, 44950,
- 78778 ]
-</pre></div>
-
-<p>Finally we calculate the characters induced from all cyclic subgroups
- and obtain the missing irreducibles by applying the LLL-algorithm to
-them.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">ic:= InducedCyclic( c, "all" );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ro:= ReducedCharacters( c, d.irreducibles, ic );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Length( ro.irreducibles );</span>
-0
-<span class="GAPprompt">gap></span> <span class="GAPinput">l:= LLL( c, ro.remainders );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Length( l.irreducibles );</span>
-13
-</pre></div>
-
-<p>The LLL returns class function objects (see Chapter <a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X7C91D0D17850E564"><span class="RefLink">72</span></a>),
- and the Dixon record works with character values lists. So we convert
-them to a list of values before feeding them in the machinery of the
-Dixon-algorithm.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">l.irreducibles[1];</span>
-Character( CharacterTable( PSL(4, 3) ),
- [ 640, -8, -8, -8, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
- 0, 0, 0, 0, E(13)^7+E(13)^8+E(13)^11, E(13)^4+E(13)^10+E(13)^12,
- E(13)^2+E(13)^5+E(13)^6, E(13)+E(13)^3+E(13)^9, 0 ] )
-<span class="GAPprompt">gap></span> <span class="GAPinput">l:=List(l.irreducibles,ValuesOfClassFunction);;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">DxIncludeIrreducibles( d, l );</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Length( d.irreducibles );</span>
-29
-<span class="GAPprompt">gap></span> <span class="GAPinput">Length( d.classes );</span>
-29
-</pre></div>
-
-<p>It turns out we have found all irreducible characters. As the last
-step, we obtain the irreducible characters and tell them to the group.
-This makes them available also to the character table.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">irrs:= DixontinI( d );;</span>
-#I Total:1 matrices,[ 2 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">SetIrr(g,irrs);</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Length(Irr(c));</span>
-29
-<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel( InfoCharacterTable, 0 );</span>
-</pre></div>
-
-<p><a id="X7C38C5067941D496" name="X7C38C5067941D496"></a></p>
-
-<h4>71.20 <span class="Heading">Constructing Character Tables from Others</span></h4>
-
-<p>The following operations take one or more character table arguments, and return a character table. This holds also for <code class="func">BrauerTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8476B25A79D7A7FC"><span class="RefLink">71.3-2</span></a>). note that the return value of <code class="func">BrauerTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8476B25A79D7A7FC"><span class="RefLink">71.3-2</span></a>) will in general not know the irreducible Brauer characters, and <strong class="pkg">GAP</strong> might be unable to compute these characters.</p>
-
-<p><em>Note</em> that whenever fusions between input and output tables
-occur in these operations, they are stored on the concerned tables, and
-the <code class="func">NamesOfFusionSources</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X7F6569D5786A9D49"><span class="RefLink">73.3-5</span></a>) values are updated.</p>
-
-<p>(The interactive construction of character tables using character
-theoretic methods and incomplete tables is not described here.) <em>Currently it is not supported and will be described in a chapter of its own when it becomes available</em>.</p>
-
-<p><a id="X7BE1572D7BBC6AC8" name="X7BE1572D7BBC6AC8"></a></p>
-
-<h5>71.20-1 CharacterTableDirectProduct</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTableDirectProduct</code>( <var class="Arg">tbl1</var>, <var class="Arg">tbl2</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>is the table of the direct product of the character tables <var class="Arg">tbl1</var> and <var class="Arg">tbl2</var>.</p>
-
-<p>The matrix of irreducibles of this table is the Kronecker product (see <code class="func">KroneckerProduct</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap24_mj.html#X8634C79E7DB22934"><span class="RefLink">24.5-8</span></a>)) of the irreducibles of <var class="Arg">tbl1</var> and <var class="Arg">tbl2</var>.</p>
-
-<p>Products of ordinary and Brauer character tables are supported.</p>
-
-<p>In general, the result will not know an underlying group, so missing
-power maps (for prime divisors of the result) and irreducibles of the
-input tables may be computed in order to construct the table of the
-direct product.</p>
-
-<p>The embeddings of the input tables into the direct product are stored, they can be fetched with <code class="func">GetFusionMap</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X8464DD23879431D9"><span class="RefLink">73.3-3</span></a>); if <var class="Arg">tbl1</var> is equal to <var class="Arg">tbl2</var> then the two embeddings are distinguished by their <code class="code">specification</code> components <code class="code">"1"</code> and <code class="code">"2"</code>, respectively.</p>
-
-<p>Analogously, the projections from the direct product onto the input tables are stored, and can be distinguished by the <code class="code">specification</code> components.</p>
-
-<p>The attribute <code class="func">FactorsOfDirectProduct</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C97CF727FBDFCAB"><span class="RefLink">71.20-2</span></a>) is set to the lists of arguments.</p>
-
-<p>The <code class="code">*</code> operator for two character tables (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CADCBC9824CB624"><span class="RefLink">71.7</span></a>) delegates to <code class="func">CharacterTableDirectProduct</code>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">c2:= CharacterTable( "Cyclic", 2 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">s3:= CharacterTable( "Symmetric", 3 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Display( CharacterTableDirectProduct( c2, s3 ) );</span>
-C2xSym(3)
-
- 2 2 2 1 2 2 1
- 3 1 . 1 1 . 1
-
- 1a 2a 3a 2b 2c 6a
- 2P 1a 1a 3a 1a 1a 3a
- 3P 1a 2a 1a 2b 2c 2b
-
-X.1 1 -1 1 1 -1 1
-X.2 2 . -1 2 . -1
-X.3 1 1 1 1 1 1
-X.4 1 -1 1 -1 1 -1
-X.5 2 . -1 -2 . 1
-X.6 1 1 1 -1 -1 -1
-</pre></div>
-
-<p><a id="X7C97CF727FBDFCAB" name="X7C97CF727FBDFCAB"></a></p>
-
-<h5>71.20-2 FactorsOfDirectProduct</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ FactorsOfDirectProduct</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>For an ordinary character table that has been constructed via <code class="func">CharacterTableDirectProduct</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7BE1572D7BBC6AC8"><span class="RefLink">71.20-1</span></a>), the value of <code class="func">FactorsOfDirectProduct</code> is the list of arguments in the <code class="func">CharacterTableDirectProduct</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7BE1572D7BBC6AC8"><span class="RefLink">71.20-1</span></a>) call.</p>
-
-<p>Note that there is no default method for <em>computing</em> the value of <code class="func">FactorsOfDirectProduct</code>.</p>
-
-<p><a id="X7C3A4E5283B240BE" name="X7C3A4E5283B240BE"></a></p>
-
-<h5>71.20-3 CharacterTableFactorGroup</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTableFactorGroup</code>( <var class="Arg">tbl</var>, <var class="Arg">classes</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>is the character table of the factor group of the ordinary character table <var class="Arg">tbl</var> by the normal closure of the classes whose positions are contained in the list <var class="Arg">classes</var>.</p>
-
-<p>The <code class="code">/</code> operator for a character table and a list of class positions (see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7CADCBC9824CB624"><span class="RefLink">71.7</span></a>) delegates to <code class="func">CharacterTableFactorGroup</code>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">s4:= CharacterTable( "Symmetric", 4 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfNormalSubgroups( s4 );</span>
-[ [ 1 ], [ 1, 3 ], [ 1, 3, 4 ], [ 1 .. 5 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CharacterTableFactorGroup( s4, [ 3 ] );</span>
-CharacterTable( "Sym(4)/[ 1, 3 ]" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">Display( f );</span>
-Sym(4)/[ 1, 3 ]
-
- 2 1 1 .
- 3 1 . 1
-
- 1a 2a 3a
- 2P 1a 1a 3a
- 3P 1a 2a 1a
-
-X.1 1 -1 1
-X.2 2 . -1
-X.3 1 1 1
-</pre></div>
-
-<p><a id="X85BE46F784B83938" name="X85BE46F784B83938"></a></p>
-
-<h5>71.20-4 CharacterTableIsoclinic</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTableIsoclinic</code>( <var class="Arg">tbl</var>[, <var class="Arg">classes</var>][, <var class="Arg">centre</var>] )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SourceOfIsoclinicTable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>If <var class="Arg">tbl</var> is the (ordinary or modular) character table of a group with the structure <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2.</mn><mi>G</mi><mn>.2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-198-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.828em; display: inline-block;" role="math" id="MathJax-Span-1174" class="math"><span style="display: inline-block; position: relative; width: 2.163em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1002.14em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1175" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1176" class="mn">2.</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1177" class="mi">G</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1178" class="mn">.2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2.</mn><mi>G</mi><mn>.2</mn></math></span></span><script id="MathJax-Element-198" type="math/tex">2.G.2</script></span> with a central subgroup <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-199-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.828em; display: inline-block;" role="math" id="MathJax-Span-1179" class="math"><span style="display: inline-block; position: relative; width: 0.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.63em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1180" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1181" class="mi">Z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.05em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script id="MathJax-Element-199" type="math/tex">Z</script></span> of order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-200-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1182" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.46em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1183" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1184" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span></span><script id="MathJax-Element-200" type="math/tex">2</script></span> or <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-201-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1185" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.45em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1186" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1187" class="mn">4</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span></span><script id="MathJax-Element-201" type="math/tex">4</script></span> and a normal subgroup <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-202-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1188" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1189" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1190" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-202" type="math/tex">N</script></span> of index <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-203-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1191" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.46em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1192" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1193" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span></span><script id="MathJax-Element-203" type="math/tex">2</script></span> that contains <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-204-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.828em; display: inline-block;" role="math" id="MathJax-Span-1194" class="math"><span style="display: inline-block; position: relative; width: 0.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.63em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1195" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1196" class="mi">Z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.05em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script id="MathJax-Element-204" type="math/tex">Z</script></span> then <code class="func">CharacterTableIsoclinic</code> returns the table of the isoclinic group in the sense of the <strong class="pkg">Atlas</strong> of Finite Groups <a href="http://www.gap-system.org/Manuals/doc/ref/chapBib_mj.html#biBCCN85">[CCNPW85, Chapter 6, Section 7]</a>. If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-205-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1197" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1198" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1199" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-205" type="math/tex">N</script></span> is not uniquely determined then the positions of the classes forming <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-206-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1200" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1201" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1202" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-206" type="math/tex">N</script></span> must be entered as list <var class="Arg">classes</var>. If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-207-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.828em; display: inline-block;" role="math" id="MathJax-Span-1203" class="math"><span style="display: inline-block; position: relative; width: 0.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.63em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1204" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1205" class="mi">Z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.05em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script id="MathJax-Element-207" type="math/tex">Z</script></span> is not unique inside <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-208-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1206" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1207" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1208" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-208" type="math/tex">N</script></span> then the positions of the classes in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-209-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.828em; display: inline-block;" role="math" id="MathJax-Span-1209" class="math"><span style="display: inline-block; position: relative; width: 0.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.63em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1210" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1211" class="mi">Z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.05em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script id="MathJax-Element-209" type="math/tex">Z</script></span> must be entered as list <var class="Arg">centre</var>; If <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-210-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.828em; display: inline-block;" role="math" id="MathJax-Span-1212" class="math"><span style="display: inline-block; position: relative; width: 0.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.63em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1213" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1214" class="mi">Z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.05em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script id="MathJax-Element-210" type="math/tex">Z</script></span> has order <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-211-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.641em; display: inline-block;" role="math" id="MathJax-Span-1215" class="math"><span style="display: inline-block; position: relative; width: 0.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1000.46em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1216" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1217" class="mn">2</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 1.004em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span></span><script id="MathJax-Element-211" type="math/tex">2</script></span> then <var class="Arg">centre</var> can be also the position of the involution in <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-212-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.828em; display: inline-block;" role="math" id="MathJax-Span-1218" class="math"><span style="display: inline-block; position: relative; width: 0.625em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.63em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1219" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1220" class="mi">Z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.05em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math></span></span><script id="MathJax-Element-212" type="math/tex">Z</script></span>.</p>
-
-<p>Note that also if <var class="Arg">tbl</var> is a Brauer table then <var class="Arg">classes</var> and <var class="Arg">centre</var> denote class numbers w.r.t. the <em>ordinary</em> character table.</p>
-
-<p>For an ordinary character table that has been constructed via <code class="func">CharacterTableIsoclinic</code>, the value of <code class="func">SourceOfIsoclinicTable</code> is the list of three arguments in the <code class="func">CharacterTableIsoclinic</code> call.</p>
-
-<p>Note that there is no default method for <em>computing</em> the value of <code class="func">SourceOfIsoclinicTable</code>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">d8:= CharacterTable( "Dihedral", 8 );</span>
-CharacterTable( "Dihedral(8)" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">nsg:= ClassPositionsOfNormalSubgroups( d8 );</span>
-[ [ 1 ], [ 1, 3 ], [ 1 .. 3 ], [ 1, 3, 4 ], [ 1, 3 .. 5 ], [ 1 .. 5 ]
- ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">iso:= CharacterTableIsoclinic( d8, nsg[3] );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Display( iso );</span>
-Isoclinic(Dihedral(8))
-
- 2 3 2 3 2 2
-
- 1a 4a 2a 4b 4c
- 2P 1a 2a 1a 2a 2a
-
-X.1 1 1 1 1 1
-X.2 1 1 1 -1 -1
-X.3 1 -1 1 1 -1
-X.4 1 -1 1 -1 1
-X.5 2 . -2 . .
-<span class="GAPprompt">gap></span> <span class="GAPinput">SourceOfIsoclinicTable( iso );</span>
-[ CharacterTable( "Dihedral(8)" ), [ 1, 2, 3 ], [ 3 ], 3 ]
-</pre></div>
-
-<p><a id="X79B75C8582426BC5" name="X79B75C8582426BC5"></a></p>
-
-<h5>71.20-5 CharacterTableWreathSymmetric</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTableWreathSymmetric</code>( <var class="Arg">tbl</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>returns the character table of the wreath product of a group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-213-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1221" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1222" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1223" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-213" type="math/tex">G</script></span> with the full symmetric group on <var class="Arg">n</var> points, where <var class="Arg">tbl</var> is the character table of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-214-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1224" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1225" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1226" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-214" type="math/tex">G</script></span>.</p>
-
-<p>The result has values for <code class="func">ClassParameters</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8333E8038308947E"><span class="RefLink">71.9-7</span></a>) and <code class="func">CharacterParameters</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8333E8038308947E"><span class="RefLink">71.9-7</span></a>)
- stored, the entries in these lists are sequences of partitions. Note
-that this parametrization prevents the principal character from being
-the first one in the list of irreducibles.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">c3:= CharacterTable( "Cyclic", 3 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">wr:= CharacterTableWreathSymmetric( c3, 2 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">Display( wr );</span>
-C3wrS2
-
- 2 1 . . 1 . 1 1 1 1
- 3 2 2 2 2 2 2 1 1 1
-
- 1a 3a 3b 3c 3d 3e 2a 6a 6b
- 2P 1a 3b 3a 3e 3d 3c 1a 3c 3e
- 3P 1a 1a 1a 1a 1a 1a 2a 2a 2a
-
-X.1 1 1 1 1 1 1 -1 -1 -1
-X.2 2 A /A B -1 /B . . .
-X.3 2 /A A /B -1 B . . .
-X.4 1 -/A -A -A 1 -/A -1 /A A
-X.5 2 -1 -1 2 -1 2 . . .
-X.6 1 -A -/A -/A 1 -A -1 A /A
-X.7 1 1 1 1 1 1 1 1 1
-X.8 1 -/A -A -A 1 -/A 1 -/A -A
-X.9 1 -A -/A -/A 1 -A 1 -A -/A
-
-A = -E(3)^2
- = (1+Sqrt(-3))/2 = 1+b3
-B = 2*E(3)
- = -1+Sqrt(-3) = 2b3
-<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterParameters( wr )[1];</span>
-[ [ 1, 1 ], [ ], [ ] ]
-</pre></div>
-
-<p><a id="X816FCD5A805F9FE8" name="X816FCD5A805F9FE8"></a></p>
-
-<h4>71.21 <span class="Heading">Sorted Character Tables</span></h4>
-
-<p><a id="X7D9C4A7F8086F671" name="X7D9C4A7F8086F671"></a></p>
-
-<h5>71.21-1 CharacterTableWithSortedCharacters</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTableWithSortedCharacters</code>( <var class="Arg">tbl</var>[, <var class="Arg">perm</var>] )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>is a character table that differs from <var class="Arg">tbl</var> only by the succession of its irreducible characters. This affects the values of the attributes <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) and <code class="func">CharacterParameters</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8333E8038308947E"><span class="RefLink">71.9-7</span></a>). Namely, these lists are permuted by the permutation <var class="Arg">perm</var>.</p>
-
-<p>If no second argument is given then a permutation is used that yields
- irreducible characters of increasing degree for the result. For the
-succession of characters in the result, see <code class="func">SortedCharacters</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87E3CF317D8E4EC7"><span class="RefLink">71.21-2</span></a>).</p>
-
-<p>The result has all those attributes and properties of <var class="Arg">tbl</var> that are stored in <code class="func">SupportedCharacterTableInfo</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7DBEF4BF87F10CD6"><span class="RefLink">71.3-4</span></a>) and do not depend on the ordering of characters.</p>
-
-<p><a id="X87E3CF317D8E4EC7" name="X87E3CF317D8E4EC7"></a></p>
-
-<h5>71.21-2 SortedCharacters</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SortedCharacters</code>( <var class="Arg">tbl</var>, <var class="Arg">chars</var>[, <var class="Arg">flag</var>] )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>is a list containing the characters <var class="Arg">chars</var>, ordered as specified by the other arguments.</p>
-
-<p>There are three possibilities to sort characters: They can be sorted according to ascending norms (<var class="Arg">flag</var> is the string <code class="code">"norm"</code>), to ascending degree (<var class="Arg">flag</var> is the string <code class="code">"degree"</code>),
- or both (no third argument is given), i.e., characters with same norm
-are sorted according to ascending degree, and characters with smaller
-norm precede those with bigger norm.</p>
-
-<p>Rational characters in the result precede other ones with same norm and/or same degree.</p>
-
-<p>The trivial character, if contained in <var class="Arg">chars</var>, will always be sorted to the first position.</p>
-
-<p><a id="X7E3DE0A47E62BE6B" name="X7E3DE0A47E62BE6B"></a></p>
-
-<h5>71.21-3 CharacterTableWithSortedClasses</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ CharacterTableWithSortedClasses</code>( <var class="Arg">tbl</var>[, <var class="Arg">flag</var>] )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>is a character table obtained by permutation of the classes of <var class="Arg">tbl</var>. If the second argument <var class="Arg">flag</var> is the string <code class="code">"centralizers"</code> then the classes of the result are sorted according to descending centralizer orders. If the second argument is the string <code class="code">"representatives"</code>
- then the classes of the result are sorted according to ascending
-representative orders. If no second argument is given then the classes
-of the result are sorted according to ascending representative orders,
-and classes with equal representative orders are sorted according to
-descending centralizer orders.</p>
-
-<p>If the second argument is a permutation then the classes of the result are sorted by application of this permutation.</p>
-
-<p>The result has all those attributes and properties of <var class="Arg">tbl</var> that are stored in <code class="func">SupportedCharacterTableInfo</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7DBEF4BF87F10CD6"><span class="RefLink">71.3-4</span></a>) and do not depend on the ordering of classes.</p>
-
-<p><a id="X82DCAAA882416E24" name="X82DCAAA882416E24"></a></p>
-
-<h5>71.21-4 SortedCharacterTable</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SortedCharacterTable</code>( <var class="Arg">tbl</var>, <var class="Arg">kernel</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SortedCharacterTable</code>( <var class="Arg">tbl</var>, <var class="Arg">normalseries</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ SortedCharacterTable</code>( <var class="Arg">tbl</var>, <var class="Arg">facttbl</var>, <var class="Arg">kernel</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>is a character table obtained on permutation of the classes and the irreducibles characters of <var class="Arg">tbl</var>.</p>
-
-<p>The first form sorts the classes at positions contained in the list <var class="Arg">kernel</var> to the beginning, and sorts all characters in <code class="code">Irr( <var class="Arg">tbl</var> )</code> such that the first characters are those that contain <var class="Arg">kernel</var> in their kernel.</p>
-
-<p>The second form does the same successively for all kernels <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mi>i</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-215-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1227" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.72em, 2.754em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1228" class="mrow"><span id="MathJax-Span-1229" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.46em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1230" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1231" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mi>i</mi></msub></math></span></span><script id="MathJax-Element-215" type="math/tex">k_i</script></span> in the list <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">normalseries</mtext></mrow><mo>=</mo><mo stretchy="false">[</mo><msub><mi>k</mi><mn>1</mn></msub><mo>,</mo><msub><mi>k</mi><mn>2</mn></msub><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><msub><mi>k</mi><mi>n</mi></msub><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-216-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 15.891em; display: inline-block;" role="math" id="MathJax-Span-1232" class="math"><span style="display: inline-block; position: relative; width: 12.212em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1012.12em, 2.945em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1233" class="mrow"><span id="MathJax-Span-1234" class="texatom"><span id="MathJax-Span-1235" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1236" class="mtext">normalseries</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1237" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1238" class="mo">[</span><span id="MathJax-Span-1239" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.46em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1240" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1241" class="mn">1</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1242" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1243" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.46em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1244" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1245" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1246" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1247" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1248" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-1249" class="msubsup"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.46em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1250" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1251" class="mi">n</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1252" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.266em; border-left: 0px solid; width: 0px; height: 1.216em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext class="MJX-tex-mathit" mathvariant="italic">normalseries</mtext></mrow><mo>=</mo><mo stretchy="false">[</mo><msub><mi>k</mi><mn>1</mn></msub><mo>,</mo><msub><mi>k</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>k</mi><mi>n</mi></msub><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-216" type="math/tex">\textit{normalseries} = [ k_1, k_2, \ldots, k_n ]</script></span> where <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mi>i</mi></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-217-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1253" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.72em, 2.754em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1254" class="mrow"><span id="MathJax-Span-1255" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.46em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1256" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1257" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mi>i</mi></msub></math></span></span><script id="MathJax-Element-217" type="math/tex">k_i</script></span> must be a sublist of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>+</mo><mn>1</mn></mrow></mrow></msub></math>" style="position: relative;" tabindex="0" id="MathJax-Element-218-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.016em; display: inline-block;" role="math" id="MathJax-Span-1258" class="math"><span style="display: inline-block; position: relative; width: 1.538em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1001.54em, 2.775em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-1259" class="mrow"><span id="MathJax-Span-1260" class="msubsup"><span style="display: inline-block; position: relative; width: 1.553em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.46em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1261" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.017em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span id="MathJax-Span-1262" class="texatom"><span id="MathJax-Span-1263" class="mrow"><span id="MathJax-Span-1264" class="texatom"><span id="MathJax-Span-1265" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1266" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1267" class="mo">+</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1268" class="mn">1</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.295em; border-left: 0px solid; width: 0px; height: 1.246em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>+</mo><mn>1</mn></mrow></mrow></msub></math></span></span><script id="MathJax-Element-218" type="math/tex">k_{{i+1}}</script></span> for <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>&#x2264;</mo><mi>i</mi><mo>&#x2264;</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></math>" style="position: relative;" tabindex="0" id="MathJax-Element-219-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.141em; display: inline-block;" role="math" id="MathJax-Span-1269" class="math"><span style="display: inline-block; position: relative; width: 5.481em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.824em, 1005.38em, 2.891em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1270" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1271" class="mn">1</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1272" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1273" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1274" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1275" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1276" class="mo">−</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1277" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.196em; border-left: 0px solid; width: 0px; height: 1.138em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script id="MathJax-Element-219" type="math/tex">1 \leq i \leq n-1</script></span>.</p>
-
-<p>The third form computes the table <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-220-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-1278" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.67em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1279" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1280" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math></span></span><script id="MathJax-Element-220" type="math/tex">F</script></span> of the factor group of <var class="Arg">tbl</var> modulo the normal subgroup formed by the classes whose positions are contained in the list <var class="Arg">kernel</var>; <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-221-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-1281" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.67em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1282" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1283" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math></span></span><script id="MathJax-Element-221" type="math/tex">F</script></span> must be permutation equivalent to the table <var class="Arg">facttbl</var>, in the sense of <code class="func">TransformingPermutationsCharacterTables</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X849731AA7EC9FA73"><span class="RefLink">71.22-4</span></a>), otherwise <code class="keyw">fail</code> is returned. The classes of <var class="Arg">tbl</var> are sorted such that the preimages of a class of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-222-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-1284" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.67em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1285" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1286" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math></span></span><script id="MathJax-Element-222" type="math/tex">F</script></span> are consecutive, and that the succession of preimages is that of <var class="Arg">facttbl</var>. The <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) value of <var class="Arg">tbl</var> is sorted as with <code class="code">SortCharTable( <var class="Arg">tbl</var>, <var class="Arg">kernel</var> )</code>.</p>
-
-<p>(<em>Note</em> that the transformation is only unique up to table automorphisms of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-223-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-1287" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.67em, 2.788em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1288" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1289" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.974em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math></span></span><script id="MathJax-Element-223" type="math/tex">F</script></span>, and this need not be unique up to table automorphisms of <var class="Arg">tbl</var>.)</p>
-
-<p>All rearrangements of classes and characters are stable, i.e., the
-relative positions of classes and characters that are not distinguished
-by any relevant property is not changed.</p>
-
-<p>The result has all those attributes and properties of <var class="Arg">tbl</var> that are stored in <code class="func">SupportedCharacterTableInfo</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7DBEF4BF87F10CD6"><span class="RefLink">71.3-4</span></a>) and do not depend on the ordering of classes and characters.</p>
-
-<p>The <code class="func">ClassPermutation</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X8099FEDC7DE03AEE"><span class="RefLink">71.21-5</span></a>) value of <var class="Arg">tbl</var> is changed if necessary, see <a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X829C4B6E83998F40"><span class="RefLink">71.5</span></a>.</p>
-
-<p><code class="func">SortedCharacterTable</code> uses <code class="func">CharacterTableWithSortedClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E3DE0A47E62BE6B"><span class="RefLink">71.21-3</span></a>) and <code class="func">CharacterTableWithSortedCharacters</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D9C4A7F8086F671"><span class="RefLink">71.21-1</span></a>).</p>
-
-<p><a id="X8099FEDC7DE03AEE" name="X8099FEDC7DE03AEE"></a></p>
-
-<h5>71.21-5 ClassPermutation</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPermutation</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>is a permutation <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-224-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1290" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.53em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1291" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1292" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.705em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ï€</mi></math></span></span><script id="MathJax-Element-224" type="math/tex">\pi</script></span> of classes of the character table <var class="Arg">tbl</var>. If it is stored then class fusions into <var class="Arg">tbl</var> that are stored on other tables must be followed by <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-225-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1293" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.53em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1294" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1295" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.705em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ï€</mi></math></span></span><script id="MathJax-Element-225" type="math/tex">\pi</script></span> in order to describe the correct fusion.</p>
-
-<p>This attribute value is bound only if <var class="Arg">tbl</var> was obtained from another table by permuting the classes, using <code class="func">CharacterTableWithSortedClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7E3DE0A47E62BE6B"><span class="RefLink">71.21-3</span></a>) or <code class="func">SortedCharacterTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X82DCAAA882416E24"><span class="RefLink">71.21-4</span></a>).</p>
-
-<p>It is necessary because the original table and the sorted table have
-the same identifier (and the same group if known), and hence the same
-fusions are valid for the two tables.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "Symmetric", 4 );</span>
-CharacterTable( "Sym(4)" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tbl );</span>
-Sym(4)
-
- 2 3 2 3 . 2
- 3 1 . . 1 .
-
- 1a 2a 2b 3a 4a
- 2P 1a 1a 1a 3a 2b
- 3P 1a 2a 2b 1a 4a
-
-X.1 1 -1 1 1 -1
-X.2 3 -1 -1 . 1
-X.3 2 . 2 -1 .
-X.4 3 1 -1 . -1
-X.5 1 1 1 1 1
-<span class="GAPprompt">gap></span> <span class="GAPinput">srt1:= CharacterTableWithSortedCharacters( tbl );</span>
-CharacterTable( "Sym(4)" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">List( Irr( srt1 ), Degree );</span>
-[ 1, 1, 2, 3, 3 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">srt2:= CharacterTableWithSortedClasses( tbl );</span>
-CharacterTable( "Sym(4)" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">SizesCentralizers( tbl );</span>
-[ 24, 4, 8, 3, 4 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">SizesCentralizers( srt2 );</span>
-[ 24, 8, 4, 3, 4 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">nsg:= ClassPositionsOfNormalSubgroups( tbl );</span>
-[ [ 1 ], [ 1, 3 ], [ 1, 3, 4 ], [ 1 .. 5 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">srt3:= SortedCharacterTable( tbl, nsg );</span>
-CharacterTable( "Sym(4)" )
-<span class="GAPprompt">gap></span> <span class="GAPinput">nsg:= ClassPositionsOfNormalSubgroups( srt3 );</span>
-[ [ 1 ], [ 1, 2 ], [ 1 .. 3 ], [ 1 .. 5 ] ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">Display( srt3 );</span>
-Sym(4)
-
- 2 3 3 . 2 2
- 3 1 . 1 . .
-
- 1a 2a 3a 2b 4a
- 2P 1a 1a 3a 1a 2a
- 3P 1a 2a 1a 2b 4a
-
-X.1 1 1 1 1 1
-X.2 1 1 1 -1 -1
-X.3 2 2 -1 . .
-X.4 3 -1 . -1 1
-X.5 3 -1 . 1 -1
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPermutation( srt3 );</span>
-(2,4,3)
-</pre></div>
-
-<p><a id="X7B0A669484470D09" name="X7B0A669484470D09"></a></p>
-
-<h4>71.22 <span class="Heading">Automorphisms and Equivalence of Character Tables</span></h4>
-
-<p><a id="X84353BB884AF0365" name="X84353BB884AF0365"></a></p>
-
-<h5>71.22-1 MatrixAutomorphisms</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ MatrixAutomorphisms</code>( <var class="Arg">mat</var>[, <var class="Arg">maps</var>, <var class="Arg">subgroup</var>] )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>For a matrix <var class="Arg">mat</var>, <code class="func">MatrixAutomorphisms</code> returns the group of those permutations of the columns of <var class="Arg">mat</var> that leave the set of rows of <var class="Arg">mat</var> invariant.</p>
-
-<p>If the arguments <var class="Arg">maps</var> and <var class="Arg">subgroup</var> are given, only the group of those permutations is constructed that additionally fix each list in the list <var class="Arg">maps</var> under pointwise action <code class="func">OnTuples</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap41_mj.html#X832CC5F87EEA4A7E"><span class="RefLink">41.2-5</span></a>), and <var class="Arg">subgroup</var> is a permutation group that is known to be a subgroup of this group of automorphisms.</p>
-
-<p>Each entry in <var class="Arg">maps</var> must be a list of same length as the rows of <var class="Arg">mat</var>. For example, if <var class="Arg">mat</var> is a list of irreducible characters of a group then the list of element orders of the conjugacy classes (see <code class="func">OrdersClassRepresentatives</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>)) may be an entry in <var class="Arg">maps</var>.</p>
-
-<p><a id="X8082DD827C673138" name="X8082DD827C673138"></a></p>
-
-<h5>71.22-2 TableAutomorphisms</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ TableAutomorphisms</code>( <var class="Arg">tbl</var>, <var class="Arg">characters</var>[, <var class="Arg">info</var>] )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p><code class="func">TableAutomorphisms</code> returns the permutation group of those matrix automorphisms (see <code class="func">MatrixAutomorphisms</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X84353BB884AF0365"><span class="RefLink">71.22-1</span></a>)) of the list <var class="Arg">characters</var> that leave the element orders (see <code class="func">OrdersClassRepresentatives</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>)) and all stored power maps (see <code class="func">ComputedPowerMaps</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap73_mj.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>)) of the character table <var class="Arg">tbl</var> invariant.</p>
-
-<p>If <var class="Arg">characters</var> is closed under Galois conjugacy
- –this is always fulfilled for the list of all irreducible characters of
- ordinary character tables– the string <code class="code">"closed"</code> may be entered as the third argument <var class="Arg">info</var>. Alternatively, a known subgroup of the table automorphisms can be entered as the third argument <var class="Arg">info</var>.</p>
-
-<p>The attribute <code class="func">AutomorphismsOfTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C2753DE8094F4BA"><span class="RefLink">71.9-4</span></a>) can be used to compute and store the table automorphisms for the case that <var class="Arg">characters</var> equals the <code class="func">Irr</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) value of <var class="Arg">tbl</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tbld8:= CharacterTable( "Dihedral", 8 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">irrd8:= Irr( tbld8 );</span>
-[ Character( CharacterTable( "Dihedral(8)" ), [ 1, 1, 1, 1, 1 ] ),
- Character( CharacterTable( "Dihedral(8)" ), [ 1, 1, 1, -1, -1 ] ),
- Character( CharacterTable( "Dihedral(8)" ), [ 1, -1, 1, 1, -1 ] ),
- Character( CharacterTable( "Dihedral(8)" ), [ 1, -1, 1, -1, 1 ] ),
- Character( CharacterTable( "Dihedral(8)" ), [ 2, 0, -2, 0, 0 ] ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">orders:= OrdersClassRepresentatives( tbld8 );</span>
-[ 1, 4, 2, 2, 2 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">MatrixAutomorphisms( irrd8 );</span>
-Group([ (4,5), (2,4) ])
-<span class="GAPprompt">gap></span> <span class="GAPinput">MatrixAutomorphisms( irrd8, [ orders ], Group( () ) );</span>
-Group([ (4,5) ])
-<span class="GAPprompt">gap></span> <span class="GAPinput">TableAutomorphisms( tbld8, irrd8 );</span>
-Group([ (4,5) ])
-</pre></div>
-
-<p><a id="X7D721E3D7AA319F5" name="X7D721E3D7AA319F5"></a></p>
-
-<h5>71.22-3 TransformingPermutations</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ TransformingPermutations</code>( <var class="Arg">mat1</var>, <var class="Arg">mat2</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">mat1</var> and <var class="Arg">mat2</var> be matrices. <code class="func">TransformingPermutations</code> tries to construct a permutation <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-226-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1296" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.53em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1297" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1298" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.705em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ï€</mi></math></span></span><script id="MathJax-Element-226" type="math/tex">\pi</script></span> that transforms the set of rows of the matrix <var class="Arg">mat1</var> to the set of rows of the matrix <var class="Arg">mat2</var> by permuting the columns.</p>
-
-<p>If such a permutation exists, a record with the components <code class="code">columns</code>, <code class="code">rows</code>, and <code class="code">group</code> is returned, otherwise <code class="keyw">fail</code>. For <code class="code">TransformingPermutations( <var class="Arg">mat1</var>, <var class="Arg">mat2</var> ) = <var class="Arg">r</var></code> <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2260;</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-227-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.891em; display: inline-block;" role="math" id="MathJax-Span-1299" class="math"><span style="display: inline-block; position: relative; width: 0.673em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.838em, 1000.63em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1300" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1301" class="mo">≠</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.188em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≠</mo></math></span></span><script id="MathJax-Element-227" type="math/tex">\neq</script></span> <code class="keyw">fail</code>, we have <code class="code"><var class="Arg">mat2</var> = Permuted( List( <var class="Arg">mat1</var>, x -> Permuted( x, <var class="Arg">r</var>.columns ) ), <var class="Arg">r</var>.rows )</code>.</p>
-
-<p><var class="Arg">r</var><code class="code">.group</code> is the group of matrix automorphisms of <var class="Arg">mat2</var> (see <code class="func">MatrixAutomorphisms</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X84353BB884AF0365"><span class="RefLink">71.22-1</span></a>)). This group stabilizes the transformation in the sense that applying any of its elements to the columns of <var class="Arg">mat2</var> preserves the set of rows of <var class="Arg">mat2</var>.</p>
-
-<p><a id="X849731AA7EC9FA73" name="X849731AA7EC9FA73"></a></p>
-
-<h5>71.22-4 TransformingPermutationsCharacterTables</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ TransformingPermutationsCharacterTables</code>( <var class="Arg">tbl1</var>, <var class="Arg">tbl2</var> )</td><td class="tdright">( operation )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">tbl1</var> and <var class="Arg">tbl2</var> be character tables. <code class="func">TransformingPermutationsCharacterTables</code> tries to construct a permutation <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-228-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1302" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.53em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1303" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1304" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.705em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ï€</mi></math></span></span><script id="MathJax-Element-228" type="math/tex">\pi</script></span> that transforms the set of rows of the matrix <code class="code">Irr( <var class="Arg">tbl1</var> )</code> to the set of rows of the matrix <code class="code">Irr( <var class="Arg">tbl2</var> )</code> by permuting the columns (see <code class="func">TransformingPermutations</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7D721E3D7AA319F5"><span class="RefLink">71.22-3</span></a>)), such that <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-229-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1305" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.53em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1306" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1307" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.705em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ï€</mi></math></span></span><script id="MathJax-Element-229" type="math/tex">\pi</script></span> transforms also the power maps and the element orders.</p>
-
-<p>If such a permutation <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-230-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1308" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.53em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1309" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1310" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.705em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ï€</mi></math></span></span><script id="MathJax-Element-230" type="math/tex">\pi</script></span> exists then a record with the components <code class="code">columns</code> (<span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-231-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1311" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.53em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1312" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1313" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.705em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ï€</mi></math></span></span><script id="MathJax-Element-231" type="math/tex">\pi</script></span>), <code class="code">rows</code> (the permutation of <code class="code">Irr( <var class="Arg">tbl1</var> )</code> corresponding to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C0;</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-232-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.703em; display: inline-block;" role="math" id="MathJax-Span-1314" class="math"><span style="display: inline-block; position: relative; width: 0.529em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.072em, 1000.53em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1315" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1316" class="mi">Ï€<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 0.705em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ï€</mi></math></span></span><script id="MathJax-Element-232" type="math/tex">\pi</script></span>), and <code class="code">group</code> (the permutation group of table automorphisms of <var class="Arg">tbl2</var>, see <code class="func">AutomorphismsOfTable</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C2753DE8094F4BA"><span class="RefLink">71.9-4</span></a>)) is returned. If no such permutation exists, <code class="keyw">fail</code> is returned.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">tblq8:= CharacterTable( "Quaternionic", 8 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">irrq8:= Irr( tblq8 );</span>
-[ Character( CharacterTable( "Q8" ), [ 1, 1, 1, 1, 1 ] ),
- Character( CharacterTable( "Q8" ), [ 1, 1, 1, -1, -1 ] ),
- Character( CharacterTable( "Q8" ), [ 1, -1, 1, 1, -1 ] ),
- Character( CharacterTable( "Q8" ), [ 1, -1, 1, -1, 1 ] ),
- Character( CharacterTable( "Q8" ), [ 2, 0, -2, 0, 0 ] ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( tblq8 );</span>
-[ 1, 4, 2, 4, 4 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutations( irrd8, irrq8 );</span>
-rec( columns := (), group := Group([ (4,5), (2,4) ]), rows := () )
-<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutationsCharacterTables( tbld8, tblq8 );</span>
-fail
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbld6:= CharacterTable( "Dihedral", 6 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbls3:= CharacterTable( "Symmetric", 3 );;</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutationsCharacterTables( tbld6, tbls3 );</span>
-rec( columns := (2,3), group := Group(()), rows := (1,3,2) )
-</pre></div>
-
-<p><a id="X8117D940835B0B47" name="X8117D940835B0B47"></a></p>
-
-<h5>71.22-5 FamiliesOfRows</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ FamiliesOfRows</code>( <var class="Arg">mat</var>, <var class="Arg">maps</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>distributes the rows of the matrix <var class="Arg">mat</var> into families, as follows. Two rows of <var class="Arg">mat</var> belong to the same family if there is a permutation of columns that maps one row to the other row. Each entry in the list <var class="Arg">maps</var> is regarded to form a family of length 1.</p>
-
-<p><code class="func">FamiliesOfRows</code> returns a record with the components</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code">famreps</code></strong></dt>
-<dd><p>the list of representatives for each family,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">permutations</code></strong></dt>
-<dd><p>the list that contains at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-233-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-1317" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1318" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1319" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-233" type="math/tex">i</script></span> a list of permutations that map the members of the family with representative <code class="code">famreps</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-234-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-1320" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.838em, 1000.87em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1321" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1322" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1323" class="mi">i</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1324" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.188em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-234" type="math/tex">[i]</script></span> to that representative,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">families</code></strong></dt>
-<dd><p>the list that contains at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-235-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-1325" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1326" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1327" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-235" type="math/tex">i</script></span> the list of positions of members of the family of representative <code class="code">famreps</code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-236-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-1328" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.838em, 1000.87em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1329" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1330" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1331" class="mi">i</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1332" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.188em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-236" type="math/tex">[i]</script></span>; (for the element <var class="Arg">maps</var><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></math>" style="position: relative;" tabindex="0" id="MathJax-Element-237-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.266em; display: inline-block;" role="math" id="MathJax-Span-1333" class="math"><span style="display: inline-block; position: relative; width: 0.962em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.838em, 1000.87em, 2.944em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1334" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1335" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1336" class="mi">i</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1337" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.265em; border-left: 0px solid; width: 0px; height: 1.188em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></math></span></span><script id="MathJax-Element-237" type="math/tex">[i]</script></span> the only member of the family will get the number <code class="code">Length( <var class="Arg">mat</var> ) + </code><span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-238-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-1338" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1339" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1340" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-238" type="math/tex">i</script></span>).</p>
-
-</dd>
-</dl>
-<p><a id="X81272CEE79F13E7B" name="X81272CEE79F13E7B"></a></p>
-
-<h4>71.23 <span class="Heading">Storing Normal Subgroup Information</span></h4>
-
-<p><a id="X7E66174C7C7A8C0C" name="X7E66174C7C7A8C0C"></a></p>
-
-<h5>71.23-1 NormalSubgroupClassesInfo</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ NormalSubgroupClassesInfo</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></tbody></table></div>
-<p>Let <var class="Arg">tbl</var> be the ordinary character table of the group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-239-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1341" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1342" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1343" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-239" type="math/tex">G</script></span>. Many computations for group characters of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-240-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1344" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1345" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1346" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-240" type="math/tex">G</script></span> involve computations in normal subgroups or factor groups of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-241-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1347" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1348" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1349" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-241" type="math/tex">G</script></span>.</p>
-
-<p>In some cases the character table <var class="Arg">tbl</var> is sufficient; for example questions about a normal subgroup <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-242-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1350" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1351" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1352" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-242" type="math/tex">N</script></span> of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-243-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1353" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1354" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1355" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-243" type="math/tex">G</script></span> can be answered if one knows the conjugacy classes that form <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-244-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1356" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1357" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1358" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-244" type="math/tex">N</script></span>, e.g., the question whether a character of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-245-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1359" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1360" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1361" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-245" type="math/tex">G</script></span> restricts irreducibly to <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-246-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1362" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1363" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1364" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-246" type="math/tex">N</script></span>. But other questions require the computation of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-247-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1365" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1366" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1367" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-247" type="math/tex">N</script></span> or even more information, like the character table of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-248-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1368" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.847em, 1000.72em, 2.803em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1369" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1370" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.082em; border-left: 0px solid; width: 0px; height: 0.993em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script id="MathJax-Element-248" type="math/tex">N</script></span>.</p>
-
-<p>In order to do these computations only once, one stores in the group a
- record with components to store normal subgroups, the corresponding
-lists of conjugacy classes, and (if necessary) the factor groups, namely</p>
-
-
-<dl>
-<dt><strong class="Mark"><code class="code">nsg</code></strong></dt>
-<dd><p>list of normal subgroups of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-249-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1371" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1372" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1373" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-249" type="math/tex">G</script></span>, may be incomplete,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">nsgclasses</code></strong></dt>
-<dd><p>at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-250-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-1374" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1375" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1376" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-250" type="math/tex">i</script></span>, the list of positions of conjugacy classes of <var class="Arg">tbl</var> forming the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-251-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-1377" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1378" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1379" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-251" type="math/tex">i</script></span>-th entry of the <code class="code">nsg</code> component,</p>
-
-</dd>
-<dt><strong class="Mark"><code class="code">nsgfactors</code></strong></dt>
-<dd><p>at position <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-252-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-1380" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1381" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1382" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-252" type="math/tex">i</script></span>, if bound, the factor group modulo the <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-253-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.391em; display: inline-block;" role="math" id="MathJax-Span-1383" class="math"><span style="display: inline-block; position: relative; width: 0.288em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.846em, 1000.27em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1384" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1385" class="mi">i</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 0.989em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script id="MathJax-Element-253" type="math/tex">i</script></span>-th entry of the <code class="code">nsg</code> component.</p>
-
-</dd>
-</dl>
-<p><code class="func">NormalSubgroupClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87E7391F7F92377C"><span class="RefLink">71.23-3</span></a>), <code class="func">FactorGroupNormalSubgroupClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X79D451F0808EB252"><span class="RefLink">71.23-4</span></a>), and <code class="func">ClassPositionsOfNormalSubgroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X7C2A87E085111090"><span class="RefLink">71.23-2</span></a>) each use these components, and they are the only functions to do so.</p>
-
-<p>So if you need information about a normal subgroup for that you know the conjugacy classes, you should get it using <code class="func">NormalSubgroupClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87E7391F7F92377C"><span class="RefLink">71.23-3</span></a>).
- If the normal subgroup was already used it is just returned, with all
-the knowledge it contains. Otherwise the normal subgroup is added to the
- lists, and will be available for the next call.</p>
-
-<p>For example, if you are dealing with kernels of characters using the <code class="func">KernelOfCharacter</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X7E0A24498710F12B"><span class="RefLink">72.8-9</span></a>) function you make use of this feature because <code class="func">KernelOfCharacter</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html#X7E0A24498710F12B"><span class="RefLink">72.8-9</span></a>) calls <code class="func">NormalSubgroupClasses</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html#X87E7391F7F92377C"><span class="RefLink">71.23-3</span></a>).</p>
-
-<p><a id="X7C2A87E085111090" name="X7C2A87E085111090"></a></p>
-
-<h5>71.23-2 ClassPositionsOfNormalSubgroup</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfNormalSubgroup</code>( <var class="Arg">tbl</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>is the list of positions of conjugacy classes of the character table <var class="Arg">tbl</var> that are contained in the normal subgroup <var class="Arg">N</var> of the underlying group of <var class="Arg">tbl</var>.</p>
-
-<p><a id="X87E7391F7F92377C" name="X87E7391F7F92377C"></a></p>
-
-<h5>71.23-3 NormalSubgroupClasses</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ NormalSubgroupClasses</code>( <var class="Arg">tbl</var>, <var class="Arg">classes</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>returns the normal subgroup of the underlying group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-254-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1386" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1387" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1388" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-254" type="math/tex">G</script></span> of the ordinary character table <var class="Arg">tbl</var> that consists of those conjugacy classes of <var class="Arg">tbl</var> whose positions are in the list <var class="Arg">classes</var>.</p>
-
-<p>If <code class="code">NormalSubgroupClassesInfo( <var class="Arg">tbl</var> ).nsg</code> does not yet contain the required normal subgroup, and if <code class="code">NormalSubgroupClassesInfo( <var class="Arg">tbl</var> ).normalSubgroups</code> is bound then the result will be identical to the group in <code class="code">NormalSubgroupClassesInfo( <var class="Arg">tbl</var> ).normalSubgroups</code>.</p>
-
-<p><a id="X79D451F0808EB252" name="X79D451F0808EB252"></a></p>
-
-<h5>71.23-4 FactorGroupNormalSubgroupClasses</h5>
-
-<div class="func"><table class="func" width="100%"><tbody><tr><td class="tdleft"><code class="func">‣ FactorGroupNormalSubgroupClasses</code>( <var class="Arg">tbl</var>, <var class="Arg">classes</var> )</td><td class="tdright">( function )</td></tr></tbody></table></div>
-<p>is the factor group of the underlying group <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-255-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1389" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1390" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1391" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-255" type="math/tex">G</script></span> of the ordinary character table <var class="Arg">tbl</var> modulo the normal subgroup of <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math>" style="position: relative;" tabindex="0" id="MathJax-Element-256-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 0.953em; display: inline-block;" role="math" id="MathJax-Span-1392" class="math"><span style="display: inline-block; position: relative; width: 0.721em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.834em, 1000.72em, 2.806em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-1393" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1394" class="mi">G</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.086em; border-left: 0px solid; width: 0px; height: 1.014em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></span></span><script id="MathJax-Element-256" type="math/tex">G</script></span> that consists of those conjugacy classes of <var class="Arg">tbl</var> whose positions are in the list <var class="Arg">classes</var>.</p>
-
-
-<div class="example"><pre><span class="GAPprompt">gap></span> <span class="GAPinput">g:= SymmetricGroup( 4 );</span>
-Sym( [ 1 .. 4 ] )
-<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( g, "S4" );</span>
-<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( g );</span>
-CharacterTable( S4 )
-<span class="GAPprompt">gap></span> <span class="GAPinput">irr:= Irr( g );</span>
-[ Character( CharacterTable( S4 ), [ 1, -1, 1, 1, -1 ] ),
- Character( CharacterTable( S4 ), [ 3, -1, -1, 0, 1 ] ),
- Character( CharacterTable( S4 ), [ 2, 0, 2, -1, 0 ] ),
- Character( CharacterTable( S4 ), [ 3, 1, -1, 0, -1 ] ),
- Character( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] ) ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">kernel:= KernelOfCharacter( irr[3] );</span>
-Group([ (1,2)(3,4), (1,3)(2,4) ])
-<span class="GAPprompt">gap></span> <span class="GAPinput">HasNormalSubgroupClassesInfo( tbl );</span>
-true
-<span class="GAPprompt">gap></span> <span class="GAPinput">NormalSubgroupClassesInfo( tbl );</span>
-rec( nsg := [ Group([ (1,2)(3,4), (1,3)(2,4) ]) ],
- nsgclasses := [ [ 1, 3 ] ], nsgfactors := [ ] )
-<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfNormalSubgroup( tbl, kernel );</span>
-[ 1, 3 ]
-<span class="GAPprompt">gap></span> <span class="GAPinput">FactorGroupNormalSubgroupClasses( tbl, [ 1, 3 ] );</span>
-Group([ f1, f2 ])
-<span class="GAPprompt">gap></span> <span class="GAPinput">NormalSubgroupClassesInfo( tbl );</span>
-rec( nsg := [ Group([ (1,2)(3,4), (1,3)(2,4) ]) ],
- nsgclasses := [ [ 1, 3 ] ], nsgfactors := [ Group([ f1, f2 ]) ] )
-</pre></div>
-
-
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+
+<p id="mathjaxlink" class="pcenter"><a href="chap60.html">[MathJax off]</a></p>
+<p><a id="X80510B5880521FDC" name="X80510B5880521FDC"></a></p>
+<div class="ChapSects"><a href="chap60_mj.html#X80510B5880521FDC">60 <span class="Heading">Abelian Number Fields</span></a>
+<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60_mj.html#X7D4E43E5799753B5">60.1 <span class="Heading">Construction of Abelian Number Fields</span></a>
+</span>
+<div class="ContSSBlock">
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X80D21D80850EFA4B">60.1-1 CyclotomicField</a></span>
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X80E5AD028143E11E">60.1-2 AbelianNumberField</a></span>
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X82F53C65802FF551">60.1-3 GaussianRationals</a></span>
+</div></div>
+<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60_mj.html#X81B5FE06781DB824">60.2 <span class="Heading">Operations for Abelian Number Fields</span></a>
+</span>
+<div class="ContSSBlock">
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X7B0AB0FB7A4136C4">60.2-1 Factors</a></span>
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X87D78F5E875F2E8A">60.2-2 IsNumberField</a></span>
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X7D202D707D5708FA">60.2-3 IsAbelianNumberField</a></span>
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X84CAE4627F0CD639">60.2-4 IsCyclotomicField</a></span>
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X87E7313D8070B9CC">60.2-5 GaloisStabilizer</a></span>
+</div></div>
+<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60_mj.html#X7D2421AC8491D2BE">60.3 <span class="Heading">Integral Bases of Abelian Number Fields</span></a>
+</span>
+<div class="ContSSBlock">
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X7F52BEA0862E06F2">60.3-1 ZumbroichBase</a></span>
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X87DB9C2C858B722A">60.3-2 LenstraBase</a></span>
+</div></div>
+<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60_mj.html#X7E4AB4B17C7BA10C">60.4 <span class="Heading">Galois Groups of Abelian Number Fields</span></a>
+</span>
+<div class="ContSSBlock">
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X7B55A90582E818F3">60.4-1 GaloisGroup</a></span>
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X8643D4B47A827D9D">60.4-2 ANFAutomorphism</a></span>
+</div></div>
+<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60_mj.html#X85E9E90D7FE877CC">60.5 <span class="Heading">Gaussians</span></a>
+</span>
+<div class="ContSSBlock">
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X80BD5EAB879F096E">60.5-1 GaussianIntegers</a></span>
+<span class="ContSS"><br /><span class="nocss"> </span><a href="chap60_mj.html#X7BFD33D27BFB7C5A">60.5-2 IsGaussianIntegers</a></span>
+</div></div>
+</div>
+
+<h3>60 <span class="Heading">Abelian Number Fields</span></h3>
+
+<p>An <em>abelian number field</em> is a field in characteristic zero that is a finite dimensional normal extension of its prime field such that the Galois group is abelian. In <strong class="pkg">GAP</strong>, one implementation of abelian number fields is given by fields of cyclotomic numbers (see Chapter <a href="chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>). Note that abelian number fields can also be constructed with the more general <code class="func">AlgebraicExtension</code> (<a href="chap67_mj.html#X7CDA90537D2BAC8A"><span class="RefLink">67.1-1</span></a>), a discussion of advantages and disadvantages can be found in <a href="chap18_mj.html#X8557FC2D7ACD6105"><span class="RefLink">18.6</span></a>. The functions described in this chapter have been developed for fields whose elements are in the filter <code class="func">IsCyclotomic</code> (<a href="chap18_mj.html#X841C425281A6F775"><span class="RefLink">18.1-3</span></a>), they may or may not work well for abelian number fields consisting of other kinds of elements.</p>
+
+<p>Throughout this chapter, <span class="SimpleMath">\(â„š_n\)</span> will denote the cyclotomic field generated by the field <span class="SimpleMath">\(â„š\)</span> of rationals together with <span class="SimpleMath">\(n\)</span>-th roots of unity.</p>
+
+<p>In <a href="chap60_mj.html#X7D4E43E5799753B5"><span class="RefLink">60.1</span></a>, constructors for abelian number fields are described, <a href="chap60_mj.html#X81B5FE06781DB824"><span class="RefLink">60.2</span></a> introduces operations for abelian number fields, <a href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a> deals with the vector space structure of abelian number fields, and <a href="chap60_mj.html#X7E4AB4B17C7BA10C"><span class="RefLink">60.4</span></a> describes field automorphisms of abelian number fields,</p>
+
+<p><a id="X7D4E43E5799753B5" name="X7D4E43E5799753B5"></a></p>
+
+<h4>60.1 <span class="Heading">Construction of Abelian Number Fields</span></h4>
+
+<p>Besides the usual construction using <code class="func">Field</code> (<a href="chap58_mj.html#X871AA7D58263E9AC"><span class="RefLink">58.1-3</span></a>) or <code class="func">DefaultField</code> (<a href="chap18_mj.html#X7FE3D5637B5485D0"><span class="RefLink">18.1-16</span></a>) (see <code class="func">DefaultField</code> (<a href="chap18_mj.html#X7FE3D5637B5485D0"><span class="RefLink">18.1-16</span></a>)), abelian number fields consisting of cyclotomics can be created with <code class="func">CyclotomicField</code> (<a href="chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) and <code class="func">AbelianNumberField</code> (<a href="chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>).</p>
+
+<p><a id="X80D21D80850EFA4B" name="X80D21D80850EFA4B"></a></p>
+
+<h5>60.1-1 CyclotomicField</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclotomicField</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclotomicField</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CF</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CF</code>( [<var class="Arg">subfield</var>, ]<var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
+<p>The first version creates the <var class="Arg">n</var>-th cyclotomic field <span class="SimpleMath">\(ℚ_n\)</span>. The second version creates the smallest cyclotomic field containing the elements in the list <var class="Arg">gens</var>. In both cases the field can be generated as an extension of a designated subfield <var class="Arg">subfield</var> (cf. <a href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p>
+
+<p><code class="func">CyclotomicField</code> can be abbreviated to <code class="func">CF</code>, this form is used also when <strong class="pkg">GAP</strong> prints cyclotomic fields.</p>
+
+<p>Fields constructed with the one argument version of <code class="func">CF</code> are stored in the global list <code class="code">CYCLOTOMIC_FIELDS</code>, so repeated calls of <code class="func">CF</code> just fetch these field objects after they have been created once.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">CyclotomicField( 5 ); CyclotomicField( [ Sqrt(3) ] );</span>
+CF(5)
+CF(12)
+<span class="GAPprompt">gap></span> <span class="GAPinput">CF( CF(3), 12 ); CF( CF(4), [ Sqrt(7) ] );</span>
+AsField( CF(3), CF(12) )
+AsField( GaussianRationals, CF(28) )
+</pre></div>
+
+<p><a id="X80E5AD028143E11E" name="X80E5AD028143E11E"></a></p>
+
+<h5>60.1-2 AbelianNumberField</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianNumberField</code>( <var class="Arg">n</var>, <var class="Arg">stab</var> )</td><td class="tdright">( function )</td></tr></table></div>
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NF</code>( <var class="Arg">n</var>, <var class="Arg">stab</var> )</td><td class="tdright">( function )</td></tr></table></div>
+<p>For a positive integer <var class="Arg">n</var> and a list <var class="Arg">stab</var> of prime residues modulo <var class="Arg">n</var>, <code class="func">AbelianNumberField</code> returns the fixed field of the group described by <var class="Arg">stab</var> (cf. <code class="func">GaloisStabilizer</code> (<a href="chap60_mj.html#X87E7313D8070B9CC"><span class="RefLink">60.2-5</span></a>)), in the <var class="Arg">n</var>-th cyclotomic field. <code class="func">AbelianNumberField</code> is mainly thought for internal use and for printing fields in a standard way; <code class="func">Field</code> (<a href="chap58_mj.html#X871AA7D58263E9AC"><span class="RefLink">58.1-3</span></a>) (cf. also <a href="chap60_mj.html#X81B5FE06781DB824"><span class="RefLink">60.2</span></a>) is probably more suitable if one knows generators of the field in question.</p>
+
+<p><code class="func">AbelianNumberField</code> can be abbreviated to <code class="func">NF</code>, this form is used also when <strong class="pkg">GAP</strong> prints abelian number fields.</p>
+
+<p>Fields constructed with <code class="func">NF</code> are stored in the global list <code class="code">ABELIAN_NUMBER_FIELDS</code>, so repeated calls of <code class="func">NF</code> just fetch these field objects after they have been created once.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">NF( 7, [ 1 ] );</span>
+CF(7)
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= NF( 7, [ 1, 2 ] ); Sqrt(-7); Sqrt(-7) in f;</span>
+NF(7,[ 1, 2, 4 ])
+E(7)+E(7)^2-E(7)^3+E(7)^4-E(7)^5-E(7)^6
+true
+</pre></div>
+
+<p><a id="X82F53C65802FF551" name="X82F53C65802FF551"></a></p>
+
+<h5>60.1-3 GaussianRationals</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GaussianRationals</code></td><td class="tdright">( global variable )</td></tr></table></div>
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGaussianRationals</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
+<p><code class="func">GaussianRationals</code> is the field <span class="SimpleMath">\(ℚ_4 = ℚ(\sqrt{{-1}})\)</span> of Gaussian rationals, as a set of cyclotomic numbers, see Chapter <a href="chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a> for basic operations. This field can also be obtained as <code class="code">CF(4)</code> (see <code class="func">CyclotomicField</code> (<a href="chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>)).</p>
+
+<p>The filter <code class="func">IsGaussianRationals</code> returns <code class="keyw">true</code> for the <strong class="pkg">GAP</strong> object <code class="func">GaussianRationals</code>, and <code class="keyw">false</code> for all other <strong class="pkg">GAP</strong> objects.</p>
+
+<p>(For details about the field of rationals, see Chapter <code class="func">Rationals</code> (<a href="chap17_mj.html#X7B6029D18570C08A"><span class="RefLink">17.1-1</span></a>).)</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">CF(4) = GaussianRationals;</span>
+true
+<span class="GAPprompt">gap></span> <span class="GAPinput">Sqrt(-1) in GaussianRationals;</span>
+true
+</pre></div>
+
+<p><a id="X81B5FE06781DB824" name="X81B5FE06781DB824"></a></p>
+
+<h4>60.2 <span class="Heading">Operations for Abelian Number Fields</span></h4>
+
+<p>For operations for elements of abelian number fields, e.g., <code class="func">Conductor</code> (<a href="chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>) or <code class="func">ComplexConjugate</code> (<a href="chap18_mj.html#X7BE001A0811CD599"><span class="RefLink">18.5-2</span></a>), see Chapter <a href="chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>.</p>
+
+<p><a id="X7B0AB0FB7A4136C4" name="X7B0AB0FB7A4136C4"></a></p>
+
+<h5>60.2-1 Factors</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Factors</code>( <var class="Arg">F</var> )</td><td class="tdright">( method )</td></tr></table></div>
+<p>Factoring of polynomials over abelian number fields consisting of cyclotomics works in principle but is not very efficient if the degree of the field extension is large.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">x:= Indeterminate( CF(5) );</span>
+x_1
+<span class="GAPprompt">gap></span> <span class="GAPinput">Factors( PolynomialRing( Rationals ), x^5-1 );</span>
+[ x_1-1, x_1^4+x_1^3+x_1^2+x_1+1 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Factors( PolynomialRing( CF(5) ), x^5-1 );</span>
+[ x_1-1, x_1+(-E(5)), x_1+(-E(5)^2), x_1+(-E(5)^3), x_1+(-E(5)^4) ]
+</pre></div>
+
+<p><a id="X87D78F5E875F2E8A" name="X87D78F5E875F2E8A"></a></p>
+
+<h5>60.2-2 IsNumberField</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNumberField</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></table></div>
+<p>returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a finite dimensional extension of a prime field in characteristic zero, and <code class="keyw">false</code> otherwise.</p>
+
+<p><a id="X7D202D707D5708FA" name="X7D202D707D5708FA"></a></p>
+
+<h5>60.2-3 IsAbelianNumberField</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAbelianNumberField</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></table></div>
+<p>returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a number field (see <code class="func">IsNumberField</code> (<a href="chap60_mj.html#X87D78F5E875F2E8A"><span class="RefLink">60.2-2</span></a>)) that is a Galois extension of the prime field, with abelian Galois group (see <code class="func">GaloisGroup</code> (<a href="chap58_mj.html#X80CAA5BA82F09ED2"><span class="RefLink">58.3-1</span></a>)).</p>
+
+<p><a id="X84CAE4627F0CD639" name="X84CAE4627F0CD639"></a></p>
+
+<h5>60.2-4 IsCyclotomicField</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCyclotomicField</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></table></div>
+<p>returns <code class="keyw">true</code> if the field <var class="Arg">F</var> is a <em>cyclotomic field</em>, i.e., an abelian number field (see <code class="func">IsAbelianNumberField</code> (<a href="chap60_mj.html#X7D202D707D5708FA"><span class="RefLink">60.2-3</span></a>)) that can be generated by roots of unity.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">IsNumberField( CF(9) ); IsAbelianNumberField( Field( [ ER(3) ] ) );</span>
+true
+true
+<span class="GAPprompt">gap></span> <span class="GAPinput">IsNumberField( GF(2) );</span>
+false
+<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( CF(9) );</span>
+true
+<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( Field( [ Sqrt(-3) ] ) );</span>
+true
+<span class="GAPprompt">gap></span> <span class="GAPinput">IsCyclotomicField( Field( [ Sqrt(3) ] ) );</span>
+false
+</pre></div>
+
+<p><a id="X87E7313D8070B9CC" name="X87E7313D8070B9CC"></a></p>
+
+<h5>60.2-5 GaloisStabilizer</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GaloisStabilizer</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
+<p>Let <var class="Arg">F</var> be an abelian number field (see <code class="func">IsAbelianNumberField</code> (<a href="chap60_mj.html#X7D202D707D5708FA"><span class="RefLink">60.2-3</span></a>)) with conductor <span class="SimpleMath">\(n\)</span>, say. (This means that the <span class="SimpleMath">\(n\)</span>-th cyclotomic field is the smallest cyclotomic field containing <var class="Arg">F</var>, see <code class="func">Conductor</code> (<a href="chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>).) <code class="func">GaloisStabilizer</code> returns the set of all those integers <span class="SimpleMath">\(k\)</span> in the range <span class="SimpleMath">\([ 1 .. n ]\)</span> such that the field automorphism induced by raising <span class="SimpleMath">\(n\)</span>-th roots of unity to the <span class="SimpleMath">\(k\)</span>-th power acts trivially on <var class="Arg">F</var>.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">r5:= Sqrt(5);</span>
+E(5)-E(5)^2-E(5)^3+E(5)^4
+<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( r5, 4 ) = r5; GaloisCyc( r5, 2 ) = r5;</span>
+true
+false
+<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisStabilizer( Field( [ r5 ] ) );</span>
+[ 1, 4 ]
+</pre></div>
+
+<p><a id="X7D2421AC8491D2BE" name="X7D2421AC8491D2BE"></a></p>
+
+<h4>60.3 <span class="Heading">Integral Bases of Abelian Number Fields</span></h4>
+
+<p>Each abelian number field is naturally a vector space over <span class="SimpleMath">\(â„š\)</span>. Moreover, if the abelian number field <span class="SimpleMath">\(F\)</span> contains the <span class="SimpleMath">\(n\)</span>-th cyclotomic field <span class="SimpleMath">\(â„š_n\)</span> then <span class="SimpleMath">\(F\)</span> is a vector space over <span class="SimpleMath">\(â„š_n\)</span>. In <strong class="pkg">GAP</strong>, each field object represents a vector space object over a certain subfield <span class="SimpleMath">\(S\)</span>, which depends on the way <span class="SimpleMath">\(F\)</span> was constructed. The subfield <span class="SimpleMath">\(S\)</span> can be accessed as the value of the attribute <code class="func">LeftActingDomain</code> (<a href="chap57_mj.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>).</p>
+
+<p>The return values of <code class="func">NF</code> (<a href="chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>) and of the one argument versions of <code class="func">CF</code> (<a href="chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) represent vector spaces over <span class="SimpleMath">\(â„š\)</span>, and the return values of the two argument version of <code class="func">CF</code> (<a href="chap60_mj.html#X80D21D80850EFA4B"><span class="RefLink">60.1-1</span></a>) represent vector spaces over the field that is given as the first argument. For an abelian number field <var class="Arg">F</var> and a subfield <var class="Arg">S</var> of <var class="Arg">F</var>, a <strong class="pkg">GAP</strong> object representing <var class="Arg">F</var> as a vector space over <var class="Arg">S</var> can be constructed using <code class="func">AsField</code> (<a href="chap58_mj.html#X7C193B7D7AFB29BE"><span class="RefLink">58.1-9</span></a>).</p>
+
+<p>Let <var class="Arg">F</var> be the cyclotomic field <span class="SimpleMath">\(ℚ_n\)</span>, represented as a vector space over the subfield <var class="Arg">S</var>. If <var class="Arg">S</var> is the cyclotomic field <span class="SimpleMath">\(ℚ_m\)</span>, with <span class="SimpleMath">\(m\)</span> a divisor of <span class="SimpleMath">\(n\)</span>, then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns the Zumbroich basis of <var class="Arg">F</var> relative to <var class="Arg">S</var>, which consists of the roots of unity <code class="code">E(<var class="Arg">n</var>)</code>^<var class="Arg">i</var> where <var class="Arg">i</var> is an element of the list <code class="code">ZumbroichBase( <var class="Arg">n</var>, <var class="Arg">m</var> )</code> (see <code class="func">ZumbroichBase</code> (<a href="chap60_mj.html#X7F52BEA0862E06F2"><span class="RefLink">60.3-1</span></a>)). If <var class="Arg">S</var> is an abelian number field that is not a cyclotomic field then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns a normal <var class="Arg">S</var>-basis of <var class="Arg">F</var>, i.e., a basis that is closed under the field automorphisms of <var class="Arg">F</var>.</p>
+
+<p>Let <var class="Arg">F</var> be the abelian number field <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stab</var> )</code>, with conductor <var class="Arg">n</var>, that is itself not a cyclotomic field, represented as a vector space over the subfield <var class="Arg">S</var>. If <var class="Arg">S</var> is the cyclotomic field <span class="SimpleMath">\(ℚ_m\)</span>, with <span class="SimpleMath">\(m\)</span> a divisor of <span class="SimpleMath">\(n\)</span>, then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns the Lenstra basis of <var class="Arg">F</var> relative to <var class="Arg">S</var> that consists of the sums of roots of unity described by <code class="code">LenstraBase( <var class="Arg">n</var>, <var class="Arg">stab</var>, <var class="Arg">stab</var>, <var class="Arg">m</var> )</code> (see <code class="func">LenstraBase</code> (<a href="chap60_mj.html#X87DB9C2C858B722A"><span class="RefLink">60.3-2</span></a>)). If <var class="Arg">S</var> is an abelian number field that is not a cyclotomic field then <code class="code">CanonicalBasis( <var class="Arg">F</var> )</code> returns a normal <var class="Arg">S</var>-basis of <var class="Arg">F</var>.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(8);; # a cycl. field over the rationals</span>
+<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
+[ 1, E(8), E(4), E(8)^3 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
+[ 0, 1, 0, 1 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AsField( CF(4), CF(8) );; # a cycl. field over a cycl. field</span>
+<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
+[ 1, E(8) ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
+[ 0, 1+E(4) ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= AsField( Field( [ Sqrt(-2) ] ), CF(8) );;</span>
+<span class="GAPprompt">gap></span> <span class="GAPinput"># a cycl. field over a non-cycl. field</span>
+<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
+[ 1/2+1/2*E(8)-1/2*E(8)^2-1/2*E(8)^3,
+ 1/2-1/2*E(8)+1/2*E(8)^2+1/2*E(8)^3 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
+[ E(8)+E(8)^3, E(8)+E(8)^3 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= Field( [ Sqrt(-2) ] ); # a non-cycl. field over the rationals</span>
+NF(8,[ 1, 3 ])
+<span class="GAPprompt">gap></span> <span class="GAPinput">b:= CanonicalBasis( f );; BasisVectors( b );</span>
+[ 1, E(8)+E(8)^3 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( b, Sqrt(-2) );</span>
+[ 0, 1 ]
+</pre></div>
+
+<p><a id="X7F52BEA0862E06F2" name="X7F52BEA0862E06F2"></a></p>
+
+<h5>60.3-1 ZumbroichBase</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZumbroichBase</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
+<p>Let <var class="Arg">n</var> and <var class="Arg">m</var> be positive integers, such that <var class="Arg">m</var> divides <var class="Arg">n</var>. <code class="func">ZumbroichBase</code> returns the set of exponents <span class="SimpleMath">\(i\)</span> for which <code class="code">E(<var class="Arg">n</var>)^</code><span class="SimpleMath">\(i\)</span> belongs to the (generalized) Zumbroich basis of the cyclotomic field <span class="SimpleMath">\(â„š_n\)</span>, viewed as a vector space over <span class="SimpleMath">\(â„š_m\)</span>.</p>
+
+<p>This basis is defined as follows. Let <span class="SimpleMath">\(P\)</span> denote the set of prime divisors of <var class="Arg">n</var>, <span class="SimpleMath">\(\textit{n} = \prod_{{p \in P}} p^{{\nu_p}}\)</span>, and <span class="SimpleMath">\(\textit{m} = \prod_{{p \in P}} p^{{\mu_p}}\)</span> with <span class="SimpleMath">\(\mu_p \leq \nu_p\)</span>. Let <span class="SimpleMath">\(e_l =\)</span> <code class="code">E</code><span class="SimpleMath">\((l)\)</span> for any positive integer <span class="SimpleMath">\(l\)</span>, and <span class="SimpleMath">\(\{ e_{{n_1}}^j \}_{{j \in J}} \otimes \{ e_{{n_2}}^k \}_{{k \in K}} = \{ e_{{n_1}}^j \cdot e_{{n_2}}^k \}_{{j \in J, k \in K}}\)</span>.</p>
+
+<p>Then the basis is</p>
+
+<p class="center">\[
+B_{{n,m}} = \bigotimes_{{p \in P}}
+ \bigotimes_{{k = \mu_p}}^{{\nu_p-1}}
+ \{ e_{{p^{{k+1}}}}^j \}_{{j \in J_{{k,p}}}}
+\]</p>
+
+<p>where <span class="SimpleMath">\(J_{{k,p}} =\)</span></p>
+
+<div class="pcenter"><table class="GAPDocTablenoborder">
+<tr>
+<td class="tdleft"><span class="SimpleMath">\(\{ 0 \}\)</span></td>
+<td class="tdcenter">;</td>
+<td class="tdleft"><span class="SimpleMath">\(k = 0, p = 2\)</span></td>
+</tr>
+<tr>
+<td class="tdleft"><span class="SimpleMath">\(\{ 0, 1 \}\)</span></td>
+<td class="tdcenter">;</td>
+<td class="tdleft"><span class="SimpleMath">\(k > 0, p = 2\)</span></td>
+</tr>
+<tr>
+<td class="tdleft"><span class="SimpleMath">\(\{ 1, \ldots, p-1 \}\)</span></td>
+<td class="tdcenter">;</td>
+<td class="tdleft"><span class="SimpleMath">\(k = 0, p \neq 2\)</span></td>
+</tr>
+<tr>
+<td class="tdleft"><span class="SimpleMath">\(\{ -(p-1)/2, \ldots, (p-1)/2 \}\)</span></td>
+<td class="tdcenter">;</td>
+<td class="tdleft"><span class="SimpleMath">\(k > 0, p \neq 2\)</span></td>
+</tr>
+</table><br /><p> </p><br />
+</div>
+
+<p><span class="SimpleMath">\(B_{{n,1}}\)</span> is equal to the basis of <span class="SimpleMath">\(ℚ_n\)</span> over the rationals which is introduced in <a href="chapBib_mj.html#biBZum89">[Zum89]</a>. Also the conversion of arbitrary sums of roots of unity into its basis representation, and the reduction to the minimal cyclotomic field are described in this thesis. (Note that the notation here is slightly different from that there.)</p>
+
+<p><span class="SimpleMath">\(B_{{n,m}}\)</span> consists of roots of unity, it is an integral basis (that is, exactly the integral elements in <span class="SimpleMath">\(ℚ_n\)</span> have integral coefficients w.r.t. <span class="SimpleMath">\(B_{{n,m}}\)</span>, cf. <code class="func">IsIntegralCyclotomic</code> (<a href="chap18_mj.html#X869750DA81EA0E67"><span class="RefLink">18.1-4</span></a>)), it is a normal basis for squarefree <span class="SimpleMath">\(n\)</span> and closed under complex conjugation for odd <span class="SimpleMath">\(n\)</span>.</p>
+
+<p><em>Note:</em> For <span class="SimpleMath">\(\textit{n} \equiv 2 \pmod 4\)</span>, we have <code class="code">ZumbroichBase(<var class="Arg">n</var>, 1) = 2 * ZumbroichBase(<var class="Arg">n</var>/2, 1)</code> and <code class="code">List( ZumbroichBase(<var class="Arg">n</var>, 1), x -> E(<var class="Arg">n</var>)^x ) = List( ZumbroichBase(<var class="Arg">n</var>/2, 1), x -> E(<var class="Arg">n</var>/2)^x )</code>.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">ZumbroichBase( 15, 1 ); ZumbroichBase( 12, 3 );</span>
+[ 1, 2, 4, 7, 8, 11, 13, 14 ]
+[ 0, 3 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">ZumbroichBase( 10, 2 ); ZumbroichBase( 32, 4 );</span>
+[ 2, 4, 6, 8 ]
+[ 0, 1, 2, 3, 4, 5, 6, 7 ]
+</pre></div>
+
+<p><a id="X87DB9C2C858B722A" name="X87DB9C2C858B722A"></a></p>
+
+<h5>60.3-2 LenstraBase</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LenstraBase</code>( <var class="Arg">n</var>, <var class="Arg">stabilizer</var>, <var class="Arg">super</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
+<p>Let <var class="Arg">n</var> and <var class="Arg">m</var> be positive integers such that <var class="Arg">m</var> divides <var class="Arg">n</var>, <var class="Arg">stabilizer</var> be a list of prime residues modulo <var class="Arg">n</var>, which describes a subfield of the <var class="Arg">n</var>-th cyclotomic field (see <code class="func">GaloisStabilizer</code> (<a href="chap60_mj.html#X87E7313D8070B9CC"><span class="RefLink">60.2-5</span></a>)), and <var class="Arg">super</var> be a list representing a supergroup of the group given by <var class="Arg">stabilizer</var>.</p>
+
+<p><code class="func">LenstraBase</code> returns a list <span class="SimpleMath">\([ b_1, b_2, \ldots, b_k ]\)</span> of lists, each <span class="SimpleMath">\(b_i\)</span> consisting of integers such that the elements <span class="SimpleMath">\(\sum_{{j \in b_i}} \)</span><code class="code">E(n)</code><span class="SimpleMath">\(^j\)</span> form a basis of the abelian number field <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stabilizer</var> )</code>, as a vector space over the <var class="Arg">m</var>-th cyclotomic field (see <code class="func">AbelianNumberField</code> (<a href="chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>)).</p>
+
+<p>This basis is an integral basis, that is, exactly the integral elements in <code class="code">NF( <var class="Arg">n</var>, <var class="Arg">stabilizer</var> )</code> have integral coefficients. (For details about this basis, see <a href="chapBib_mj.html#biBBre97">[Bre97]</a>.)</p>
+
+<p>If possible then the result is chosen such that the group described by <var class="Arg">super</var> acts on it, consistently with the action of <var class="Arg">stabilizer</var>, i.e., each orbit of <var class="Arg">super</var> is a union of orbits of <var class="Arg">stabilizer</var>. (A usual case is <var class="Arg">super</var><code class="code"> = </code><var class="Arg">stabilizer</var>, so there is no additional condition.</p>
+
+<p><em>Note:</em> The <span class="SimpleMath">\(b_i\)</span> are in general not sets, since for <code class="code"><var class="Arg">stabilizer</var> = <var class="Arg">super</var></code>, the first entry is always an element of <code class="code">ZumbroichBase( <var class="Arg">n</var>, <var class="Arg">m</var> )</code>; this property is used by <code class="func">NF</code> (<a href="chap60_mj.html#X80E5AD028143E11E"><span class="RefLink">60.1-2</span></a>) and <code class="func">Coefficients</code> (<a href="chap61_mj.html#X80B32F667BF6AFD8"><span class="RefLink">61.6-3</span></a>) (see <a href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p>
+
+<p><var class="Arg">stabilizer</var> must not contain the stabilizer of a proper cyclotomic subfield of the <var class="Arg">n</var>-th cyclotomic field, i.e., the result must describe a basis for a field with conductor <var class="Arg">n</var>.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 24, [ 1, 19 ], [ 1, 19 ], 1 );</span>
+[ [ 1, 19 ], [ 8 ], [ 11, 17 ], [ 16 ] ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 24, [ 1, 19 ], [ 1, 5, 19, 23 ], 1 );</span>
+[ [ 1, 19 ], [ 5, 23 ], [ 8 ], [ 16 ] ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">LenstraBase( 15, [ 1, 4 ], PrimeResidues( 15 ), 1 );</span>
+[ [ 1, 4 ], [ 2, 8 ], [ 7, 13 ], [ 11, 14 ] ]
+</pre></div>
+
+<p>The first two results describe two bases of the field <span class="SimpleMath">\(â„š_3(\sqrt{{6}})\)</span>, the third result describes a normal basis of <span class="SimpleMath">\(â„š_3(\sqrt{{5}})\)</span>.</p>
+
+<p><a id="X7E4AB4B17C7BA10C" name="X7E4AB4B17C7BA10C"></a></p>
+
+<h4>60.4 <span class="Heading">Galois Groups of Abelian Number Fields</span></h4>
+
+<p>The field automorphisms of the cyclotomic field <span class="SimpleMath">\(ℚ_n\)</span> (see Chapter <a href="chap18_mj.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>) are given by the linear maps <span class="SimpleMath">\(*k\)</span> on <span class="SimpleMath">\(ℚ_n\)</span> that are defined by <code class="code">E</code><span class="SimpleMath">\((n)^{{*k}} = \)</span><code class="code">E</code><span class="SimpleMath">\((n)^k\)</span>, where <span class="SimpleMath">\(1 \leq k < n\)</span> and <code class="code">Gcd</code><span class="SimpleMath">\(( n, k ) = 1\)</span> hold (see <code class="func">GaloisCyc</code> (<a href="chap18_mj.html#X79EE9097783128C4"><span class="RefLink">18.5-1</span></a>)). Note that this action is <em>not</em> equal to exponentiation of cyclotomics, i.e., for general cyclotomics <span class="SimpleMath">\(z\)</span>, <span class="SimpleMath">\(z^{{*k}}\)</span> is different from <span class="SimpleMath">\(z^k\)</span>.</p>
+
+<p>(In <strong class="pkg">GAP</strong>, the image of a cyclotomic <span class="SimpleMath">\(z\)</span> under <span class="SimpleMath">\(*k\)</span> can be computed as <code class="code">GaloisCyc( </code><span class="SimpleMath">\(z, k\)</span><code class="code"> )</code>.)</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">( E(5) + E(5)^4 )^2; GaloisCyc( E(5) + E(5)^4, 2 );</span>
+-2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
+E(5)^2+E(5)^3
+</pre></div>
+
+<p>For <code class="code">Gcd</code><span class="SimpleMath">\(( n, k ) \neq 1\)</span>, the map <code class="code">E</code><span class="SimpleMath">\((n) \mapsto\)</span> <code class="code">E</code><span class="SimpleMath">\((n)^k\)</span> does <em>not</em> define a field automorphism of <span class="SimpleMath">\(â„š_n\)</span> but only a <span class="SimpleMath">\(â„š\)</span>-linear map.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 );</span>
+2
+-6
+</pre></div>
+
+<p><a id="X7B55A90582E818F3" name="X7B55A90582E818F3"></a></p>
+
+<h5>60.4-1 GaloisGroup</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GaloisGroup</code>( <var class="Arg">F</var> )</td><td class="tdright">( method )</td></tr></table></div>
+<p>The Galois group <span class="SimpleMath">\(Gal( ℚ_n, ℚ )\)</span> of the field extension <span class="SimpleMath">\(ℚ_n / ℚ\)</span> is isomorphic to the group <span class="SimpleMath">\((ℤ / n ℤ)^{*}\)</span> of prime residues modulo <span class="SimpleMath">\(n\)</span>, via the isomorphism <span class="SimpleMath">\((ℤ / n ℤ)^{*} \rightarrow Gal( ℚ_n, ℚ )\)</span> that is defined by <span class="SimpleMath">\(k + n ℤ \mapsto ( z \mapsto z^{*k} )\)</span>.</p>
+
+<p>The Galois group of the field extension <span class="SimpleMath">\(ℚ_n / L\)</span> with any abelian number field <span class="SimpleMath">\(L \subseteq ℚ_n\)</span> is simply the factor group of <span class="SimpleMath">\(Gal( ℚ_n, ℚ )\)</span> modulo the stabilizer of <span class="SimpleMath">\(L\)</span>, and the Galois group of <span class="SimpleMath">\(L / L'\)</span>, with <span class="SimpleMath">\(L'\)</span> an abelian number field contained in <span class="SimpleMath">\(L\)</span>, is the subgroup in this group that stabilizes <span class="SimpleMath">\(L'\)</span>. These groups are easily described in terms of <span class="SimpleMath">\((ℤ / n ℤ)^{*}\)</span>. Generators of <span class="SimpleMath">\((ℤ / n ℤ)^{*}\)</span> can be computed using <code class="func">GeneratorsPrimeResidues</code> (<a href="chap15_mj.html#X7D191CF67E5018BE"><span class="RefLink">15.2-4</span></a>).</p>
+
+<p>In <strong class="pkg">GAP</strong>, a field extension <span class="SimpleMath">\(L / L'\)</span> is given by the field object <span class="SimpleMath">\(L\)</span> with <code class="func">LeftActingDomain</code> (<a href="chap57_mj.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>) value <span class="SimpleMath">\(L'\)</span> (see <a href="chap60_mj.html#X7D2421AC8491D2BE"><span class="RefLink">60.3</span></a>).</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(15);</span>
+CF(15)
+<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GaloisGroup( f );</span>
+<group with 2 generators>
+<span class="GAPprompt">gap></span> <span class="GAPinput">Size( g ); IsCyclic( g ); IsAbelian( g );</span>
+8
+false
+true
+<span class="GAPprompt">gap></span> <span class="GAPinput">Action( g, NormalBase( f ), OnPoints );</span>
+Group([ (1,6)(2,4)(3,8)(5,7), (1,4,3,7)(2,8,5,6) ])
+</pre></div>
+
+<p>The following example shows Galois groups of a cyclotomic field and of a proper subfield that is not a cyclotomic field.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">gens1:= GeneratorsOfGroup( GaloisGroup( CF(5) ) );</span>
+[ ANFAutomorphism( CF(5), 2 ) ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">gens2:= GeneratorsOfGroup( GaloisGroup( Field( Sqrt(5) ) ) );</span>
+[ ANFAutomorphism( NF(5,[ 1, 4 ]), 2 ) ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Order( gens1[1] ); Order( gens2[1] );</span>
+4
+2
+<span class="GAPprompt">gap></span> <span class="GAPinput">Sqrt(5)^gens1[1] = Sqrt(5)^gens2[1];</span>
+true
+</pre></div>
+
+<p>The following example shows the Galois group of a cyclotomic field over a non-cyclotomic field.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GaloisGroup( AsField( Field( [ Sqrt(5) ] ), CF(5) ) );</span>
+<group with 1 generators>
+<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( g );</span>
+[ ANFAutomorphism( AsField( NF(5,[ 1, 4 ]), CF(5) ), 4 ) ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">x:= last[1];; x^2;</span>
+IdentityMapping( AsField( NF(5,[ 1, 4 ]), CF(5) ) )
+</pre></div>
+
+<p><a id="X8643D4B47A827D9D" name="X8643D4B47A827D9D"></a></p>
+
+<h5>60.4-2 ANFAutomorphism</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ANFAutomorphism</code>( <var class="Arg">F</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
+<p>Let <var class="Arg">F</var> be an abelian number field and <var class="Arg">k</var> be an integer that is coprime to the conductor (see <code class="func">Conductor</code> (<a href="chap18_mj.html#X815D6EC57CBA9827"><span class="RefLink">18.1-7</span></a>)) of <var class="Arg">F</var>. Then <code class="func">ANFAutomorphism</code> returns the automorphism of <var class="Arg">F</var> that is defined as the linear extension of the map that raises each root of unity in <var class="Arg">F</var> to its <var class="Arg">k</var>-th power.</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CF(25);</span>
+CF(25)
+<span class="GAPprompt">gap></span> <span class="GAPinput">alpha:= ANFAutomorphism( f, 2 );</span>
+ANFAutomorphism( CF(25), 2 )
+<span class="GAPprompt">gap></span> <span class="GAPinput">alpha^2;</span>
+ANFAutomorphism( CF(25), 4 )
+<span class="GAPprompt">gap></span> <span class="GAPinput">Order( alpha );</span>
+20
+<span class="GAPprompt">gap></span> <span class="GAPinput">E(5)^alpha;</span>
+E(5)^2
+</pre></div>
+
+<p><a id="X85E9E90D7FE877CC" name="X85E9E90D7FE877CC"></a></p>
+
+<h4>60.5 <span class="Heading">Gaussians</span></h4>
+
+<p><a id="X80BD5EAB879F096E" name="X80BD5EAB879F096E"></a></p>
+
+<h5>60.5-1 GaussianIntegers</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GaussianIntegers</code></td><td class="tdright">( global variable )</td></tr></table></div>
+<p><code class="func">GaussianIntegers</code> is the ring <span class="SimpleMath">\(ℤ[\sqrt{{-1}}]\)</span> of Gaussian integers. This is a subring of the cyclotomic field <code class="func">GaussianRationals</code> (<a href="chap60_mj.html#X82F53C65802FF551"><span class="RefLink">60.1-3</span></a>).</p>
+
+<p><a id="X7BFD33D27BFB7C5A" name="X7BFD33D27BFB7C5A"></a></p>
+
+<h5>60.5-2 IsGaussianIntegers</h5>
+
+<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGaussianIntegers</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
+<p>is the defining category for the domain <code class="func">GaussianIntegers</code> (<a href="chap60_mj.html#X80BD5EAB879F096E"><span class="RefLink">60.5-1</span></a>).</p>
+
+
+<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap59_mj.html">[Previous Chapter]</a> <a href="chap61_mj.html">[Next Chapter]</a> </div>
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+
+<hr />
+<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
+</body>
+</html>
diff --git a/export/fromwebpage/http:_www.gap-system.org_Manuals_pkg_GAPDoc-1.5.1_example_chap2.html b/export/fromwebpage/http:_www.gap-system.org_Manuals_pkg_GAPDoc-1.5.1_example_chap2.html
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--- /dev/null
+++ b/export/fromwebpage/http:_www.gap-system.org_Manuals_pkg_GAPDoc-1.5.1_example_chap2.html
@@ -0,0 +1,373 @@
+<?xml version="1.0" encoding="UTF-8"?>
+
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+
+<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
+<head>
+<script type="text/javascript"
+ src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
+</script>
+<title>GAP (GAPDoc Example) - Chapter 2: Other Markup</title>
+<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
+<meta name="generator" content="GAPDoc2HTML" />
+<link rel="stylesheet" type="text/css" href="manual.css" />
+<script src="manual.js" type="text/javascript"></script>
+<script type="text/javascript">overwriteStyle();</script>
+</head>
+<body class="chap2" onload="jscontent()">
+
+
+<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chapA_mj.html">A</a> <a href="chapB_mj.html">B</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>
+
+<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap1_mj.html">[Previous Chapter]</a> <a href="chapA_mj.html">[Next Chapter]</a> </div>
+
+<p id="mathjaxlink" class="pcenter"><a href="chap2.html">[MathJax off]</a></p>
+<p><a id="X82793A7E7A3F2A54" name="X82793A7E7A3F2A54"></a></p>
+<div class="ChapSects"><a href="chap2_mj.html#X82793A7E7A3F2A54">2 <span class="Heading">Other Markup</span></a>
+<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7A480B9A795EF264">2.1 <span class="Heading">Various types of text</span></a>
+</span>
+</div>
+<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7AA5BF0279938BE0">2.2 <span class="Heading">Formulae</span></a>
+</span>
+</div>
+<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X833C410D85CF96A4">2.3 <span class="Heading">Crossreferencing</span></a>
+</span>
+</div>
+<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7F10E951789D6EDF">2.4 <span class="Heading">Lists and Tables</span></a>
+</span>
+</div>
+<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X83A355E68485D6D1">2.5 <span class="Heading">Entities and Special Characters</span></a>
+</span>
+</div>
+</div>
+
+<h3>2 <span class="Heading">Other Markup</span></h3>
+
+<p><a id="X7A480B9A795EF264" name="X7A480B9A795EF264"></a></p>
+
+<h4>2.1 <span class="Heading">Various types of text</span></h4>
+
+<p>[→ <a href="chapB_mj.html#X82A731CA83FB9ADD"><span class="RefLink">B.5</span></a>]</p>
+
+<p>In this section we present examples for all the various types of text that are possible in <strong class="pkg">GAPDoc</strong>:</p>
+
+
+<ul>
+<li><p><em>This</em> is <em>emphasized</em>.</p>
+
+</li>
+<li><p><em>Keywords</em> are typeset like <code class="keyw">this</code> and <code class="keyw">that</code>.</p>
+
+</li>
+<li><p><em>Arguments</em> of functions have an element. They look like this: <var class="Arg">x</var> and <var class="Arg">y</var>.</p>
+
+</li>
+<li><p><em>Code</em> can be written with the Code element: <code class="code">if x = y then Print("Equal"); fi;</code> or <code class="code">while true do Print("Hello"); od;</code>.</p>
+
+</li>
+<li><p><em>Filenames</em> have their own element: <code class="file">/usr/local/ca/gap4r2</code> or <code class="file">pkg/xgap/doc</code>.</p>
+
+</li>
+<li><p><em>Buttons</em>, <em>menus</em>, <em>menu entries</em>, and such things are also supported: <strong class="button">OK</strong> or <strong class="button">Cancel</strong>.</p>
+
+</li>
+<li><p><em>Packages</em> are typeset like this: <strong class="pkg">Small Groups Library</strong></p>
+
+</li>
+<li><p><em>Quoted</em> text: "This is a text in quotes."</p>
+
+</li>
+</ul>
+<p><em>Paragraphs</em> are separated by the empty <code class="code">Par</code> or <code class="code">P</code> element.</p>
+
+<p><em>Alternatives</em> for different output formats: This is other than LaTeX output, namely: <b>HTML</b> output.</p>
+
+<p>There are also three elements to typeset "verbatim-like" text. (→ <a href="chapB_mj.html#X824BD70087820CF0"><span class="RefLink">B.6</span></a>)</p>
+
+<p>The first is a <em>Listing</em>:</p>
+
+
+<div class="example"><pre>
+Sieve := function(n)
+ # Returns the primes less than n
+ local l,p,i;
+ l := [1..n]; Unbind(l[1]);
+ p := 2;
+ while p^2 <= n do
+ if IsBound(l[p]) then
+ i := 2 * p;
+ while i <= n do Unbind(l[i]); i := i + p; od;
+ fi;
+ p := p + 1;
+ od;
+ return Compacted(l);
+end;
+</pre></div>
+
+<p>Here is a <em>Log</em> of a <strong class="pkg">GAP</strong> session using this function:</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">Sieve(100);</span>
+[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
+ 67, 71, 73, 79, 83, 89, 97 ]
+<span class="GAPprompt">gap></span> <span class="GAPinput">Length(last);</span>
+25
+</pre></div>
+
+<p>Here is a <strong class="pkg">GAP</strong> <em>Example</em> session that is automatically tested:</p>
+
+
+<div class="example"><pre>
+<span class="GAPprompt">gap></span> <span class="GAPinput">s := Size(CharacterTable("M"));</span>
+808017424794512875886459904961710757005754368000000000
+<span class="GAPprompt">gap></span> <span class="GAPinput">s < 10^53; </span>
+false
+<span class="GAPprompt">gap></span> <span class="GAPinput">s < 10^54;</span>
+true
+</pre></div>
+
+<p><a id="X7AA5BF0279938BE0" name="X7AA5BF0279938BE0"></a></p>
+
+<h4>2.2 <span class="Heading">Formulae</span></h4>
+
+<p>[→ <a href="chapB_mj.html#X8516FAA07A95BBB5"><span class="RefLink">B.7</span></a>]</p>
+
+<p>There are three types of formulae.</p>
+
+<p>The first is the <em>normal math mode</em> of LaTeX: <span class="SimpleMath">\(b_i \cdot b_j = \sum_{k=1}^d h_{ijk} b_k\)</span>. Then there are <em>displayed formulae</em>:</p>
+
+<p class="center">\[
+ \Longrightarrow \quad \left(\sum_{i=1}^d x_i b_i \right) \cdot
+ \left(\sum_{j=1}^d y_j b_j \right) =
+ \sum_{k=1}^d \left( \sum_{i,j} x_i y_j h_{ijk} \right) b_k
+\]</p>
+
+<p>If possible, use the <code class="code">Alt</code> element to specify a better readable text version of such a formula as in the following example:</p>
+
+
+<pre class="normal">
+
+ d d d
+ ----- ----- ----- -----
+ \ \ \ \
+ ==> ( ) x_i b_i )( ) y_j b_j ) = ) ( ) x_i y_j h_ijk ) b_k
+ / / / /
+ ----- ----- ----- -----
+ i = 1 j = 1 k = 1 i,j
+
+</pre>
+
+<p>For small formulae without "difficult" parts use the <code class="code">M</code> element: <span class="SimpleMath">\(b_i\)</span>, <span class="SimpleMath">\(x^2\)</span>, <span class="SimpleMath">\(x^2 + 2x + 1 = (x + 1)^2\)</span>. Note that here whitespace matters for text (or HTML) output.</p>
+
+<p>Here are two formulae containing less than characters which are special characters for XML: <span class="SimpleMath">\(a < b < c < d\)</span> and <span class="SimpleMath">\(e < f\)</span>.</p>
+
+<p>Using the <code class="code">Mode</code> attribute of a <code class="code">Display</code> element formulae like</p>
+
+<p class="center">\[a \longrightarrow a \bmod m\prime\]</p>
+
+<p>can also be displayed nicely in text and HTML output.</p>
+
+<p><a id="X833C410D85CF96A4" name="X833C410D85CF96A4"></a></p>
+
+<h4>2.3 <span class="Heading">Crossreferencing</span></h4>
+
+<p>[→ <a href="chapB_mj.html#X7D19CF4782309661"><span class="RefLink">B.8</span></a>]</p>
+
+<p>In this section we demonstrate various references to parts of this document. Here is a reference to this section: <a href="chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">2.3</span></a>. Here is a reference to chapter <a href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a>, to appendix <a href="chapA_mj.html#X7B53252784137533"><span class="RefLink">A</span></a>, and to subsection <a href="chap1_mj.html#X7E193BD379F58A4C"><span class="RefLink">1.1-1</span></a>.</p>
+
+<p>We distinguish among others references to functions (see <code class="func">f</code> (<a href="chap1_mj.html#X7FA1D0937FA1D093"><span class="RefLink">1.2-1</span></a>)), to methods with tricky name (see <code class="func">\^\{\}\[\]\<\&</code> (<a href="chap1_mj.html#X822B5C487B29E799"><span class="RefLink">1.2-2</span></a>)), to operations (see <code class="func">MyOperation</code> (<a href="chap1_mj.html#X7D33C2597988F481"><span class="RefLink">1.2-3</span></a>)), to methods (see <code class="func">MyOperation</code> (<a href="chap1_mj.html#X783DCD4E826289D4"><span class="RefLink">1.2-4</span></a>) or <code class="func">MyOperation</code> (<a href="chap1_mj.html#X7A5F4A287D06988C"><span class="RefLink">1.2-5</span></a>)), to filters (see <code class="func">IsBla</code> (<a href="chap1_mj.html#X82954B687D2DF3C2"><span class="RefLink">1.2-6</span></a>)), to properties (see <code class="func">IsBlubb</code> (<a href="chap1_mj.html#X80C364DD7C919CCE"><span class="RefLink">1.2-7</span></a>)), to attributes (see <code class="func">NumberBlobbs</code> (<a href="chap1_mj.html#X8052A45E7F9F054C"><span class="RefLink">1.2-8</span></a>)), to variables (<code class="func">AllBlibbs</code> (<a href="chap1_mj.html#X7C00E05A7DDEF003"><span class="RefLink">1.2-9</span></a>)), to families (see <code class="func">BlibbsFamily</code> (<a href="chap1_mj.html#X7CBC935A8142E374"><span class="RefLink">1.2-10</span></a>)), and to info classes (see <code class="func">InfoBlibbs</code> (<a href="chap1_mj.html#X84D7D77378AD030A"><span class="RefLink">1.2-11</span></a>)).</p>
+
+<p>There are also references to labels: see <a href="chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">here</span></a>, to other books: see <a href="../../pkg/GAPDoc-1.5.1/doc/chap3_mj.html#X7B76F6F786521F6B"><span class="RefLink">GAPDoc: What is a DTD?</span></a> or <code class="func">IsSubgroup</code> (<a href="../../../doc/ref/chap40_mj.html#X7839D8927E778334"><span class="RefLink">Reference: IsSubgroup</span></a>) in the <strong class="pkg">GAP</strong> reference manual.</p>
+
+<p>References to sections come in two styles: <a href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a> or <a href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink"><span class="Heading">Sectioning Elements</span></span></a>.</p>
+
+<p>Another type of cross referencing is bibliography. Here is a citation: <a href="chapBib_mj.html#biBCR1">[CR81, (5.22)]</a> is an interesting lemma.</p>
+
+<p>There are also URLs:</p>
+
+<p><span class="URL"><a href="http://www.math.rwth-aachen.de/">http://www.math.rwth-aachen.de/</a></span></p>
+
+<p>Email addresses have a special element: <span class="URL"><a href="mailto:Frank.Luebeck@Math.RWTH-Aachen.De">Frank.Luebeck@Math.RWTH-Aachen.De</a></span></p>
+
+<p>and Homepages another one: <span class="URL"><a href="http://www-groups.mcs.st-and.ac.uk/~neunhoef/">http://www-groups.mcs.st-and.ac.uk/~neunhoef/</a></span></p>
+
+<p>And here is a link to the <span class="URL"><a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/gap/EDIM/index.html#ARCHS"><strong class="pkg">EDIM</strong> archives</a></span>.</p>
+
+<p>One can generate index entries as follows (look up the words "TeX-UserGroup", "RWTH", "Aachen, Hauptbahnhof", and "<strong class="pkg">GAP</strong>, <strong class="pkg">GAPDoc</strong>").</p>
+
+<p><a id="X7F10E951789D6EDF" name="X7F10E951789D6EDF"></a></p>
+
+<h4>2.4 <span class="Heading">Lists and Tables</span></h4>
+
+<p>[→ <a href="chapB_mj.html#X7BB822947F626E1A"><span class="RefLink">B.9</span></a>]</p>
+
+<p>There are</p>
+
+
+<ul>
+<li><p>lists</p>
+
+</li>
+<li><p>enumerations, and</p>
+
+</li>
+<li><p>tables</p>
+
+</li>
+</ul>
+<p>or:</p>
+
+<ol>
+<li><p>lists</p>
+
+</li>
+<li><p>enumerations, and</p>
+
+</li>
+<li><p>tables</p>
+
+</li>
+</ol>
+<p>or with marks:</p>
+
+
+<dl>
+<dt><strong class="Mark">lists:</strong></dt>
+<dd><p>not numbered</p>
+
+</dd>
+<dt><strong class="Mark">enumerations:</strong></dt>
+<dd><p>numbered</p>
+
+</dd>
+<dt><strong class="Mark">tables:</strong></dt>
+<dd><p>two-dimensional</p>
+
+</dd>
+</dl>
+<p>Lists can also be nested:</p>
+
+<ol>
+<li><ol>
+<li><p>first item of inner enumeration</p>
+
+</li>
+<li><p>second item of inner enumeration</p>
+
+</li>
+</ol>
+</li>
+<li>
+<ul>
+<li><p>first item of inner list</p>
+
+</li>
+<li><p>second item of inner list</p>
+
+</li>
+</ul>
+</li>
+</ol>
+<p>Here is a <em>table</em>:</p>
+
+<div class="pcenter"><table class="GAPDocTable">
+<caption class="GAPDocTable"><b>Table: </b>Prices</caption>
+<tr>
+<td class="tdright">Object</td>
+<td class="tdcenter">Price</td>
+<td class="tdleft">available</td>
+</tr>
+<tr>
+<td class="tdright">Shoe</td>
+<td class="tdcenter">$1,00</td>
+<td class="tdleft">there</td>
+</tr>
+<tr>
+<td class="tdright">Hat</td>
+<td class="tdcenter">$2,00</td>
+<td class="tdleft">not there</td>
+</tr>
+</table><br /><p> </p><br />
+</div>
+
+<p><a id="X83A355E68485D6D1" name="X83A355E68485D6D1"></a></p>
+
+<h4>2.5 <span class="Heading">Entities and Special Characters</span></h4>
+
+<p>[→ <a href="chapB_mj.html#X80B478CD7E584F6F"><span class="RefLink">B.10</span></a>]</p>
+
+<p>Here is a table of special characters, the first two are special for XML and must be typed in by entities in <strong class="pkg">GAPDoc</strong> documents. The other characters are special for LaTeX but in <strong class="pkg">GAPDoc</strong> they can be typed directly.</p>
+
+<div class="pcenter"><table class="GAPDocTable">
+<caption class="GAPDocTable"><b>Table: </b>Special characters in character data</caption>
+<tr>
+<td class="tdcenter"><code class="code">&</code></td>
+<td class="tdcenter"><code class="code"><</code></td>
+<td class="tdcenter"><code class="code">></code></td>
+<td class="tdcenter"><code class="code">#</code></td>
+<td class="tdcenter"><code class="code">$</code></td>
+<td class="tdcenter"><code class="code">%</code></td>
+<td class="tdcenter"><code class="code">~</code></td>
+<td class="tdcenter"><code class="code">\</code></td>
+<td class="tdcenter"><code class="code">{</code></td>
+<td class="tdcenter"><code class="code">}</code></td>
+<td class="tdcenter"><code class="code">_</code></td>
+<td class="tdcenter"><code class="code">^</code></td>
+<td class="tdcenter"><code class="code">Â </code></td>
+</tr>
+</table><br /><p> </p><br />
+</div>
+
+<p>And here are the predefined entities in <strong class="pkg">GAPDoc</strong>:</p>
+
+<div class="pcenter"><table class="GAPDocTable">
+<caption class="GAPDocTable"><b>Table: </b>Predefined Entities in the <strong class="pkg">GAPDoc</strong> system</caption>
+<tr>
+<td class="tdleft"><code class="code">&GAP;</code></td>
+<td class="tdleft"><strong class="pkg">GAP</strong></td>
+</tr>
+<tr>
+<td class="tdleft"><code class="code">&GAPDoc;</code></td>
+<td class="tdleft"><strong class="pkg">GAPDoc</strong></td>
+</tr>
+<tr>
+<td class="tdleft"><code class="code">&TeX;</code></td>
+<td class="tdleft">TeX</td>
+</tr>
+<tr>
+<td class="tdleft"><code class="code">&LaTeX;</code></td>
+<td class="tdleft">LaTeX</td>
+</tr>
+<tr>
+<td class="tdleft"><code class="code">&BibTeX;</code></td>
+<td class="tdleft">BibTeX</td>
+</tr>
+<tr>
+<td class="tdleft"><code class="code">&MeatAxe;</code></td>
+<td class="tdleft"><strong class="pkg">MeatAxe</strong></td>
+</tr>
+<tr>
+<td class="tdleft"><code class="code">&XGAP;</code></td>
+<td class="tdleft"><strong class="pkg">XGAP</strong></td>
+</tr>
+<tr>
+<td class="tdleft"><code class="code">&copyright;</code></td>
+<td class="tdleft">©</td>
+</tr>
+</table><br /><p> </p><br />
+</div>
+
+<p>And some more for mathematical symbols: â„‚, ℤ, â„•, â„™, â„š, â„, â„.</p>
+
+
+<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap1_mj.html">[Previous Chapter]</a> <a href="chapA_mj.html">[Next Chapter]</a> </div>
+
+
+<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chapA_mj.html">A</a> <a href="chapB_mj.html">B</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>
+
+<hr />
+<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
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