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+<!--?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="GAP%20%28GAPDoc%20Example%29%20-%20Chapter%202:%20Other%20Markup_files/MathJax.js">
+</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="GAP%20%28GAPDoc%20Example%29%20-%20Chapter%202:%20Other%20Markup_files/manual.css">
+<script src="GAP%20%28GAPDoc%20Example%29%20-%20Chapter%202:%20Other%20Markup_files/manual.js" type="text/javascript"></script>
+<script type="text/javascript">overwriteStyle();</script>
+<style type="text/css">.MathJax_Hover_Frame {border-radius: .25em; -webkit-border-radius: .25em; -moz-border-radius: .25em; -khtml-border-radius: .25em; box-shadow: 0px 0px 15px #83A; -webkit-box-shadow: 0px 0px 15px #83A; -moz-box-shadow: 0px 0px 15px #83A; -khtml-box-shadow: 0px 0px 15px #83A; border: 1px solid #A6D ! important; display: inline-block; position: absolute}
+.MathJax_Menu_Button .MathJax_Hover_Arrow {position: absolute; cursor: pointer; display: inline-block; border: 2px solid #AAA; border-radius: 4px; -webkit-border-radius: 4px; -moz-border-radius: 4px; -khtml-border-radius: 4px; font-family: 'Courier New',Courier; font-size: 9px; color: #F0F0F0}
+.MathJax_Menu_Button .MathJax_Hover_Arrow span {display: block; background-color: #AAA; border: 1px solid; border-radius: 3px; line-height: 0; padding: 4px}
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+.MathJax_Hover_Arrow:hover span {background-color: #CCC!important}
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+#MathJax_About.MathJax_MousePost {outline: none}
+.MathJax_Menu {position: absolute; background-color: white; color: black; width: auto; padding: 5px 0px; border: 1px solid #CCCCCC; margin: 0; cursor: default; font: menu; text-align: left; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; z-index: 201; border-radius: 5px; -webkit-border-radius: 5px; -moz-border-radius: 5px; -khtml-border-radius: 5px; box-shadow: 0px 10px 20px #808080; -webkit-box-shadow: 0px 10px 20px #808080; -moz-box-shadow: 0px 10px 20px #808080; -khtml-box-shadow: 0px 10px 20px #808080; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
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+.MathJax_MenuActive .MathJax_MenuArrow {color: white}
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+.MathJax_MenuCheck {position: absolute; left: .7em}
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+.MathJax_MenuLabel {padding: 1px 2em 3px 1.33em; font-style: italic}
+.MathJax_MenuRule {border-top: 1px solid #DDDDDD; margin: 4px 3px}
+.MathJax_MenuDisabled {color: GrayText}
+.MathJax_MenuActive {background-color: #606872; color: white}
+.MathJax_MenuDisabled:focus, .MathJax_MenuLabel:focus {background-color: #E8E8E8}
+.MathJax_ContextMenu:focus {outline: none}
+.MathJax_ContextMenu .MathJax_MenuItem:focus {outline: none}
+#MathJax_AboutClose {top: .2em; right: .2em}
+.MathJax_Menu .MathJax_MenuClose {top: -10px; left: -10px}
+.MathJax_MenuClose {position: absolute; cursor: pointer; display: inline-block; border: 2px solid #AAA; border-radius: 18px; -webkit-border-radius: 18px; -moz-border-radius: 18px; -khtml-border-radius: 18px; font-family: 'Courier New',Courier; font-size: 24px; color: #F0F0F0}
+.MathJax_MenuClose span {display: block; background-color: #AAA; border: 1.5px solid; border-radius: 18px; -webkit-border-radius: 18px; -moz-border-radius: 18px; -khtml-border-radius: 18px; line-height: 0; padding: 8px 0 6px}
+.MathJax_MenuClose:hover {color: white!important; border: 2px solid #CCC!important}
+.MathJax_MenuClose:hover span {background-color: #CCC!important}
+.MathJax_MenuClose:hover:focus {outline: none}
+</style><style type="text/css">.MathJax_Preview .MJXf-math {color: inherit!important}
+</style><style type="text/css">.MJX_Assistive_MathML {position: absolute!important; top: 0; left: 0; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0 0 0!important; border: 0!important; height: 1px!important; width: 1px!important; overflow: hidden!important; display: block!important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none}
+.MJX_Assistive_MathML.MJX_Assistive_MathML_Block {width: 100%!important}
+</style><style type="text/css">#MathJax_Zoom {position: absolute; background-color: #F0F0F0; overflow: auto; display: block; z-index: 301; padding: .5em; border: 1px solid black; margin: 0; font-weight: normal; font-style: normal; text-align: left; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; -webkit-box-sizing: content-box; -moz-box-sizing: content-box; box-sizing: content-box; box-shadow: 5px 5px 15px #AAAAAA; -webkit-box-shadow: 5px 5px 15px #AAAAAA; -moz-box-shadow: 5px 5px 15px #AAAAAA; -khtml-box-shadow: 5px 5px 15px #AAAAAA; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
+#MathJax_ZoomOverlay {position: absolute; left: 0; top: 0; z-index: 300; display: inline-block; width: 100%; height: 100%; border: 0; padding: 0; margin: 0; background-color: white; opacity: 0; filter: alpha(opacity=0)}
+#MathJax_ZoomFrame {position: relative; display: inline-block; height: 0; width: 0}
+#MathJax_ZoomEventTrap {position: absolute; left: 0; top: 0; z-index: 302; display: inline-block; border: 0; padding: 0; margin: 0; background-color: white; opacity: 0; filter: alpha(opacity=0)}
+</style><style type="text/css">.MathJax_Preview {color: #888}
+#MathJax_Message {position: fixed; left: 1px; bottom: 2px; background-color: #E6E6E6; border: 1px solid #959595; margin: 0px; padding: 2px 8px; z-index: 102; color: black; font-size: 80%; width: auto; white-space: nowrap}
+#MathJax_MSIE_Frame {position: absolute; top: 0; left: 0; width: 0px; z-index: 101; border: 0px; margin: 0px; padding: 0px}
+.MathJax_Error {color: #CC0000; font-style: italic}
+</style><style type="text/css">.MJXp-script {font-size: .8em}
+.MJXp-right {-webkit-transform-origin: right; -moz-transform-origin: right; -ms-transform-origin: right; -o-transform-origin: right; transform-origin: right}
+.MJXp-bold {font-weight: bold}
+.MJXp-italic {font-style: italic}
+.MJXp-scr {font-family: MathJax_Script,'Times New Roman',Times,STIXGeneral,serif}
+.MJXp-frak {font-family: MathJax_Fraktur,'Times New Roman',Times,STIXGeneral,serif}
+.MJXp-sf {font-family: MathJax_SansSerif,'Times New Roman',Times,STIXGeneral,serif}
+.MJXp-cal {font-family: MathJax_Caligraphic,'Times New Roman',Times,STIXGeneral,serif}
+.MJXp-mono {font-family: MathJax_Typewriter,'Times New Roman',Times,STIXGeneral,serif}
+.MJXp-largeop {font-size: 150%}
+.MJXp-largeop.MJXp-int {vertical-align: -.2em}
+.MJXp-math {display: inline-block; line-height: 1.2; text-indent: 0; font-family: 'Times New Roman',Times,STIXGeneral,serif; white-space: nowrap; border-collapse: collapse}
+.MJXp-display {display: block; text-align: center; margin: 1em 0}
+.MJXp-math span {display: inline-block}
+.MJXp-box {display: block!important; text-align: center}
+.MJXp-box:after {content: " "}
+.MJXp-rule {display: block!important; margin-top: .1em}
+.MJXp-char {display: block!important}
+.MJXp-mo {margin: 0 .15em}
+.MJXp-mfrac {margin: 0 .125em; vertical-align: .25em}
+.MJXp-denom {display: inline-table!important; width: 100%}
+.MJXp-denom > * {display: table-row!important}
+.MJXp-surd {vertical-align: top}
+.MJXp-surd > * {display: block!important}
+.MJXp-script-box > * {display: table!important; height: 50%}
+.MJXp-script-box > * > * {display: table-cell!important; vertical-align: top}
+.MJXp-script-box > *:last-child > * {vertical-align: bottom}
+.MJXp-script-box > * > * > * {display: block!important}
+.MJXp-mphantom {visibility: hidden}
+.MJXp-munderover {display: inline-table!important}
+.MJXp-over {display: inline-block!important; text-align: center}
+.MJXp-over > * {display: block!important}
+.MJXp-munderover > * {display: table-row!important}
+.MJXp-mtable {vertical-align: .25em; margin: 0 .125em}
+.MJXp-mtable > * {display: inline-table!important; vertical-align: middle}
+.MJXp-mtr {display: table-row!important}
+.MJXp-mtd {display: table-cell!important; text-align: center; padding: .5em 0 0 .5em}
+.MJXp-mtr > .MJXp-mtd:first-child {padding-left: 0}
+.MJXp-mtr:first-child > .MJXp-mtd {padding-top: 0}
+.MJXp-mlabeledtr {display: table-row!important}
+.MJXp-mlabeledtr > .MJXp-mtd:first-child {padding-left: 0}
+.MJXp-mlabeledtr:first-child > .MJXp-mtd {padding-top: 0}
+.MJXp-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 1px 3px; font-style: normal; font-size: 90%}
+.MJXp-scale0 {-webkit-transform: scaleX(.0); -moz-transform: scaleX(.0); -ms-transform: scaleX(.0); -o-transform: scaleX(.0); transform: scaleX(.0)}
+.MJXp-scale1 {-webkit-transform: scaleX(.1); -moz-transform: scaleX(.1); -ms-transform: scaleX(.1); -o-transform: scaleX(.1); transform: scaleX(.1)}
+.MJXp-scale2 {-webkit-transform: scaleX(.2); -moz-transform: scaleX(.2); -ms-transform: scaleX(.2); -o-transform: scaleX(.2); transform: scaleX(.2)}
+.MJXp-scale3 {-webkit-transform: scaleX(.3); -moz-transform: scaleX(.3); -ms-transform: scaleX(.3); -o-transform: scaleX(.3); transform: scaleX(.3)}
+.MJXp-scale4 {-webkit-transform: scaleX(.4); -moz-transform: scaleX(.4); -ms-transform: scaleX(.4); -o-transform: scaleX(.4); transform: scaleX(.4)}
+.MJXp-scale5 {-webkit-transform: scaleX(.5); -moz-transform: scaleX(.5); -ms-transform: scaleX(.5); -o-transform: scaleX(.5); transform: scaleX(.5)}
+.MJXp-scale6 {-webkit-transform: scaleX(.6); -moz-transform: scaleX(.6); -ms-transform: scaleX(.6); -o-transform: scaleX(.6); transform: scaleX(.6)}
+.MJXp-scale7 {-webkit-transform: scaleX(.7); -moz-transform: scaleX(.7); -ms-transform: scaleX(.7); -o-transform: scaleX(.7); transform: scaleX(.7)}
+.MJXp-scale8 {-webkit-transform: scaleX(.8); -moz-transform: scaleX(.8); -ms-transform: scaleX(.8); -o-transform: scaleX(.8); transform: scaleX(.8)}
+.MJXp-scale9 {-webkit-transform: scaleX(.9); -moz-transform: scaleX(.9); -ms-transform: scaleX(.9); -o-transform: scaleX(.9); transform: scaleX(.9)}
+.MathJax_PHTML .noError {vertical-align: ; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid}
+</style><style type="text/css">.MathJax_Display {text-align: center; margin: 1em 0em; position: relative; display: block!important; text-indent: 0; max-width: none; max-height: none; min-width: 0; min-height: 0; width: 100%}
+.MathJax .merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 1px 3px; font-style: normal; font-size: 90%}
+.MathJax .MJX-monospace {font-family: monospace}
+.MathJax .MJX-sans-serif {font-family: sans-serif}
+#MathJax_Tooltip {background-color: InfoBackground; color: InfoText; border: 1px solid black; box-shadow: 2px 2px 5px #AAAAAA; -webkit-box-shadow: 2px 2px 5px #AAAAAA; -moz-box-shadow: 2px 2px 5px #AAAAAA; -khtml-box-shadow: 2px 2px 5px #AAAAAA; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true'); padding: 3px 4px; z-index: 401; position: absolute; left: 0; top: 0; width: auto; height: auto; display: none}
+.MathJax {display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 100%; font-size-adjust: none; text-indent: 0; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; padding: 0; margin: 0}
+.MathJax:focus, body :focus .MathJax {display: inline-table}
+.MathJax img, .MathJax nobr, .MathJax a {border: 0; padding: 0; margin: 0; max-width: none; max-height: none; min-width: 0; min-height: 0; vertical-align: 0; line-height: normal; text-decoration: none}
+img.MathJax_strut {border: 0!important; padding: 0!important; margin: 0!important; vertical-align: 0!important}
+.MathJax span {display: inline; position: static; border: 0; padding: 0; margin: 0; vertical-align: 0; line-height: normal; text-decoration: none}
+.MathJax nobr {white-space: nowrap!important}
+.MathJax img {display: inline!important; float: none!important}
+.MathJax * {transition: none; -webkit-transition: none; -moz-transition: none; -ms-transition: none; -o-transition: none}
+.MathJax_Processing {visibility: hidden; position: fixed; width: 0; height: 0; overflow: hidden}
+.MathJax_Processed {display: none!important}
+.MathJax_ExBox {display: block!important; overflow: hidden; width: 1px; height: 60ex; min-height: 0; max-height: none}
+.MathJax .MathJax_EmBox {display: block!important; overflow: hidden; width: 1px; height: 60em; min-height: 0; max-height: none}
+.MathJax .MathJax_HitBox {cursor: text; background: white; opacity: 0; filter: alpha(opacity=0)}
+.MathJax .MathJax_HitBox * {filter: none; opacity: 1; background: transparent}
+#MathJax_Tooltip * {filter: none; opacity: 1; background: transparent}
+@font-face {font-family: MathJax_Blank; src: url('about:blank')}
+.MathJax .noError {vertical-align: ; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid}
+</style></head>
+<body class="chap2" onload="jscontent()"><div style="visibility: hidden; overflow: hidden; position: absolute; top: 0px; height: 1px; width: auto; padding: 0px; border: 0px none; margin: 0px; text-align: left; text-indent: 0px; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal;"><div id="MathJax_Hidden"></div></div><div style="display: none;" id="MathJax_Message"></div>
+
+
+<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>
+
+<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>
+
+<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>
+
+<h4>2.1 <span class="Heading">Various types of text</span></h4>
+
+<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>
+
+<hr>
+<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/%7EFrank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
+
+
+<div style="position: absolute; width: 0px; height: 0px; overflow: hidden; padding: 0px; border: 0px none; margin: 0px;"><div style="position: absolute; visibility: hidden; top: 0px; left: 0px; width: auto; padding: 0px; border: 0px none; margin: 0px; white-space: nowrap; text-align: left; text-indent: 0px; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; font-size: 40px; font-weight: normal; font-style: normal; font-size-adjust: none; font-family: STIXSizeOneSym,sans-serif;" id="MathJax_Font_Test"></div></div></body></html>
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+.MathJax_ContextMenu .MathJax_MenuItem:focus {outline: none}
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+<title>GAP (Wedderga) - Chapter 9: The basic theory behind Wedderga</title>
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+<meta name="generator" content="GAPDoc2HTML">
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+<body class="chap9" onload="jscontent()"><div style="visibility: hidden; overflow: hidden; position: absolute; top: 0px; height: 1px; width: auto; padding: 0px; border: 0px none; margin: 0px; text-align: left; text-indent: 0px; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal;"><div id="MathJax_Hidden"></div></div><div style="display: none;" id="MathJax_Message"></div>
+
+
+<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap0_mj.html">Top</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap1_mj.html">1</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap2_mj.html">2</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap3_mj.html">3</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap4_mj.html">4</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap5_mj.html">5</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap6_mj.html">6</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap7_mj.html">7</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap8_mj.html">8</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html">9</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html">Bib</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapInd_mj.html">Ind</a> </div>
+
+<div class="chlinkprevnexttop"> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap0_mj.html">[Top of Book]</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap0_mj.html#contents">[Contents]</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap8_mj.html">[Previous Chapter]</a> <a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chapBib_mj.html">[Next Chapter]</a> </div>
+
+<p id="mathjaxlink" class="pcenter"><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9.html">[MathJax off]</a><a href="http://www.gap-system.org/Manuals/pkg/wedderga/doc/chooser.html?BACK=http://www.gap-system.org/Manuals/pkg/wedderga/doc/chap9_mj.html"> [Style]</a></p>
+<p><a id="X8487E7AD7D6B2EA5" name="X8487E7AD7D6B2EA5"></a></p>
+<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>
+</span>
+</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#X7FDD93FB79ADCC91">9.2 <span class="Heading">Semisimple group algebras</span></a>
+</span>
+</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>
+</span>
+</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#X87B6505C7C2EE054">9.4 <span class="Heading">Characters and primitive central idempotents</span></a>
+</span>
+</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#X7A24D5407F72C633">9.5 <span class="Heading">Central simple algebras and Brauer equivalence</span></a>
+</span>
+</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#X7FB21779832CE1CB">9.6 <span class="Heading">Crossed Products</span></a>
+</span>
+</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#X828C42CD86AF605F">9.7 <span class="Heading">Cyclic Crossed Products</span></a>
+</span>
+</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#X7869E2A48784C232">9.8 <span class="Heading">Abelian Crossed Products</span></a>
+</span>
+</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#X80BABE5078A29793">9.9 <span class="Heading">Classical crossed products</span></a>
+</span>
+</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#X84C98BB8859BBEE2">9.10 <span class="Heading">Cyclic Algebras</span></a>
+</span>
+</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#X8099A8C784255672">9.11 <span class="Heading">Cyclotomic algebras</span></a>
+</span>
+</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#X84A142407B7565E0">9.12 <span class="Heading">Numerical description of cyclotomic algebras</span></a>
+</span>
+</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#X8310E96086509397">9.13 <span class="Heading">Idempotents given by subgroups</span></a>
+</span>
+</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#X7D518BAB80EDE190">9.14 <span class="Heading">Shoda pairs of a group</span></a>
+</span>
+</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#X7E3479527BAE5B9E">9.15 <span class="Heading">Strong Shoda pairs of a group</span></a>
+</span>
+</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#X84C694978557EFE5">9.16 <span class="Heading">Strongly monomial characters and strongly monomial groups</span></a>
+</span>
+</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#X800D8C5087D79DC8">9.17 <span class="Heading">Cyclotomic Classes and Strong Shoda Pairs</span></a>
+</span>
+</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#X803562E087325AF6">9.18 <span class="Heading">Theory for Local Schur Index and Division Algebra Part Calculations</span></a>
+</span>
+</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#X7B18AF347AE68020">9.19 <span class="Heading"> Obtaining Algebras with structure constants as terms of the Wedderburn decomposition </span></a>
+</span>
+</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#X8472ACCF802EC188">9.20 <span class="Heading">A complete set of orthogonal primitive idempotents</span></a>
+</span>
+</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#X856D7975810BF987">9.21 <span class="Heading">Applications to coding theory</span></a>
+</span>
+</div>
+</div>
+
+<h3>9 <span class="Heading">The basic theory behind <strong class="pkg">Wedderga</strong></span></h3>
+
+<p>In this chapter we describe the theory that is behind the algorithms used by <strong class="pkg">Wedderga</strong>.</p>
+
+<p>All the rings considered in this chapter are associative and have an identity.</p>
+
+<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>
+
+<p><a id="X84C98BB8859BBEE2" name="X84C98BB8859BBEE2"></a></p>
+
+<h4>9.10 <span class="Heading">Cyclic Algebras</span></h4>
+
+<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>
+
+<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>
+
+<p><a id="X84A142407B7565E0" name="X84A142407B7565E0"></a></p>
+
+<h4>9.12 <span class="Heading">Numerical description of cyclotomic algebras</span></h4>
+
+<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>
+
+<p><a id="X7D518BAB80EDE190" name="X7D518BAB80EDE190"></a></p>
+
+<h4>9.14 <span class="Heading">Shoda pairs of a group</span></h4>
+
+<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>
+
+<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>
+
+<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>
+
+<p><a id="X7E3479527BAE5B9E" name="X7E3479527BAE5B9E"></a></p>
+
+<h4>9.15 <span class="Heading">Strong Shoda pairs of a group</span></h4>
+
+<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>
+
+<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>
+
+<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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+<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html">Top</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap1_mj.html">1</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap2_mj.html">2</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap3_mj.html">3</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap4_mj.html">4</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap5_mj.html">5</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html">6</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap7_mj.html">7</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap8_mj.html">8</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap9_mj.html">9</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap10_mj.html">10</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap11_mj.html">11</a> <a 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href="http://www.gap-system.org/Manuals/doc/ref/chap38_mj.html">38</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html">39</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap40_mj.html">40</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap41_mj.html">41</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap42_mj.html">42</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap43_mj.html">43</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap44_mj.html">44</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap45_mj.html">45</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap46_mj.html">46</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap47_mj.html">47</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap48_mj.html">48</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap49_mj.html">49</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap50_mj.html">50</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap51_mj.html">51</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap52_mj.html">52</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap53_mj.html">53</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap54_mj.html">54</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap55_mj.html">55</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap56_mj.html">56</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap57_mj.html">57</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html">58</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap59_mj.html">59</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html">60</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap61_mj.html">61</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap62_mj.html">62</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap63_mj.html">63</a> <a 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>
+
+<div class="chlinkprevnexttop"> <a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html">[Top of Book]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html#contents">[Contents]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap17_mj.html">[Previous Chapter]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap19_mj.html">[Next Chapter]</a> </div>
+
+<p id="mathjaxlink" class="pcenter"><a href="http://www.gap-system.org/Manuals/doc/ref/chap18.html">[MathJax off]</a><a href="http://www.gap-system.org/Manuals/doc/ref/chooser.html?BACK=http://www.gap-system.org/Manuals/doc/ref/chap18_mj.html"> [Style]</a></p>
+<p><a id="X7DFC03C187DE4841" name="X7DFC03C187DE4841"></a></p>
+<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>
+</span>
+<div class="ContSSBlock">
+<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>
+</div></div>
+<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>
+<div class="ContSSBlock">
+<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>
+</div></div>
+<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>
+</span>
+</div>
+<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>
+</span>
+<div class="ContSSBlock">
+<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>
+</span>
+<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>
+</span>
+<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>
+</span>
+<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>
+</div></div>
+<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>
+<div class="ContSSBlock">
+<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>
+</div></div>
+<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>
+</span>
+<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>
+</div></div>
+</div>
+
+<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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+<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html">Top</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap1_mj.html">1</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap2_mj.html">2</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap3_mj.html">3</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap4_mj.html">4</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap5_mj.html">5</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html">6</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap7_mj.html">7</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap8_mj.html">8</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap9_mj.html">9</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap10_mj.html">10</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap11_mj.html">11</a> <a 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href="http://www.gap-system.org/Manuals/doc/ref/chap51_mj.html">51</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap52_mj.html">52</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap53_mj.html">53</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap54_mj.html">54</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap55_mj.html">55</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap56_mj.html">56</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap57_mj.html">57</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html">58</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap59_mj.html">59</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html">60</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap61_mj.html">61</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap62_mj.html">62</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap63_mj.html">63</a> <a 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>
+
+<div class="chlinkprevnexttop"> <a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html">[Top of Book]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html#contents">[Contents]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap59_mj.html">[Previous Chapter]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap61_mj.html">[Next Chapter]</a> </div>
+
+<p id="mathjaxlink" class="pcenter"><a href="http://www.gap-system.org/Manuals/doc/ref/chap60.html">[MathJax off]</a><a href="http://www.gap-system.org/Manuals/doc/ref/chooser.html?BACK=http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html"> [Style]</a></p>
+<p><a id="X80510B5880521FDC" name="X80510B5880521FDC"></a></p>
+<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>
+
+<p><a id="X7B55A90582E818F3" name="X7B55A90582E818F3"></a></p>
+
+<h5>60.4-1 GaloisGroup</h5>
+
+<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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+<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html">Top</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap1_mj.html">1</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap2_mj.html">2</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap3_mj.html">3</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap4_mj.html">4</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap5_mj.html">5</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap6_mj.html">6</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap7_mj.html">7</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap8_mj.html">8</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap9_mj.html">9</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap10_mj.html">10</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap11_mj.html">11</a> <a 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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>
+
+<div class="chlinkprevnexttop"> <a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html">[Top of Book]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html#contents">[Contents]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap70_mj.html">[Previous Chapter]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html">[Next Chapter]</a> </div>
+
+<p id="mathjaxlink" class="pcenter"><a href="http://www.gap-system.org/Manuals/doc/ref/chap71.html">[MathJax off]</a><a href="http://www.gap-system.org/Manuals/doc/ref/chooser.html?BACK=http://www.gap-system.org/Manuals/doc/ref/chap71_mj.html"> [Style]</a></p>
+<p><a id="X7B7A9EE881E01C10" name="X7B7A9EE881E01C10"></a></p>
+<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>
+</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#X7F8AB7CB7A46002F">71.2 <span class="Heading">History of Character Theory Stuff in GAP</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#X8701174D86B586AF">71.3 <span class="Heading">Creating 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#X7FCA7A7A822BDA33">71.3-1 <span class="Heading">CharacterTable</span></a>
+</span>
+<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>
+</span>
+<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>
+</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#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>
+
+<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>
+
+
+<div class="chlinkprevnextbot"> <a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html">[Top of Book]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap0_mj.html#contents">[Contents]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap70_mj.html">[Previous Chapter]</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap72_mj.html">[Next Chapter]</a> </div>
+
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href="http://www.gap-system.org/Manuals/doc/ref/chap38_mj.html">38</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap39_mj.html">39</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap40_mj.html">40</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap41_mj.html">41</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap42_mj.html">42</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap43_mj.html">43</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap44_mj.html">44</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap45_mj.html">45</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap46_mj.html">46</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap47_mj.html">47</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap48_mj.html">48</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap49_mj.html">49</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap50_mj.html">50</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap51_mj.html">51</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap52_mj.html">52</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap53_mj.html">53</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap54_mj.html">54</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap55_mj.html">55</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap56_mj.html">56</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap57_mj.html">57</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap58_mj.html">58</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap59_mj.html">59</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap60_mj.html">60</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap61_mj.html">61</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap62_mj.html">62</a> <a href="http://www.gap-system.org/Manuals/doc/ref/chap63_mj.html">63</a> <a 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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