<html lang="en"><head>
<title>Centraliser</title>
<!-- firefox doesn't allow fonts from a different parent folder so fonts (cross-domain fonts)
must be a subfolder from here -->
<link rel="stylesheet" href="katex.min.css">
<script src="../katex.min.js"></script>
<script src="myrender.js"></script>
<script src="timer.js"></script>
<script src="replacements.js"></script>
<script src="copy-tex.min.js"></script>
<link rel="stylesheet" href="copy-tex.min.css">
<style>
<style>
/* optional change maths font size */
/* optional change maths font size */
.katex { font-size: 0.9em !important;
line-height: 2.2 !important;
}
body {
margin-top:0px;
margin-left:120px;
margin-right:200px;
margin-bottom:0px;
font-size: 1.1em;
}
.eqnum {
float: right;
margin-right: 1rem;
margin-top: -2.7em;
}
.halmos {
float: right;
margin-right: 1rem;
}
.onright {
float: right;
margin-right: 1rem;
font-size: 0.875em;
color: blue;
}
.onleft {
float: left;
margin-left: 0rem;
margin-top: 0em;
font-size: 0.875em;
}
adj {
float: left;
margin-top: -2.3cm;
}
</style>
<script>
function morecommands(){
document.body.innerHTML = document.body.innerHTML.replace(/\\Stab/g, ' \\text\{Stab\}');
document.body.innerHTML = document.body.innerHTML.replace(/\\Orb/g, ' \\text\{Orb\}');
newcommand();
}
</script>
</head>
<body onload="morecommands();myonload();">
<br>
<span id="time_taken" class="onright">Time taken by KaTeX to render formulæ : 8 ms</span>
<span class="onleft"><a href="centraliser.pdf">PDF version</a></span>
<br><br><h4>Theorem </h4><em>
If <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span></span></span></span></span> is an element of <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo>=</mo><msub><mi>S</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">G=S_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathnormal">G</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83333em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.151392em;"><span class="" style="top: -2.55em; margin-left: -0.05764em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> then <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">C_G(g)=\langle g\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07153em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">⟩</span></span></span></span></span> if and only if the cycles of <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span></span></span></span></span> are of unequal coprime lengths.<br>
Alternatively, <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">C_G(g)=\langle g\rangle</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07153em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">⟩</span></span></span></span></span> if and only if <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span></span></span></span></span> has cycles of coprime length with at most one 1-cycle.
</em><br><br>
<em>Proof. </em>
Since powers of <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span></span></span></span></span> centralise <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span></span></span></span></span> it follows that <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>g</mi><mo stretchy="false">⟩</mo><mo>⊆</mo><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\langle g\rangle\subseteq C_G(g)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">⟩</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07153em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)</span></span></span></span></span> so
<div class="maths"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo stretchy="false">⟩</mo>  <mo>⟺</mo>  <mo stretchy="false">∣</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo stretchy="false">⟩</mo><mo stretchy="false">∣</mo><mo>=</mo><mo stretchy="false">∣</mo><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo>  <mo>⟺</mo>  <mo stretchy="false">∣</mo><mi>g</mi><mo stretchy="false">∣</mo><mo>=</mo><mo stretchy="false">∣</mo><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
C_G(g)=\langle g\rangle\iff \lvert\langle g\rangle\rvert=\lvert C_G(g)\rvert\iff\lvert g\rvert=\lvert C_G(g)\rvert
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.5em; vertical-align: -0.5em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1em;"><span class="" style="top: -3.16em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07153em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">⟩</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">∣⟨</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">⟩∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07153em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">∣</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07153em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></div>
<span class="eqnum">(1)</span>
Let <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.68333em; vertical-align: 0em;"></span><span class="mord mathnormal">G</span></span></span></span></span> act on itself by conjugation. Then <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>g</mi><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>x</mi>  <mo>⟺</mo>  <mi>x</mi><mi>g</mi><mo>=</mo><mi>g</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">xgx^{-1} = x\iff xg=gx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.00855em; vertical-align: -0.19444em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">xg</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.814108em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.549em; vertical-align: -0.024em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">xg</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathnormal">gx</span></span></span></span></span> so
<div class="maths"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>Stab</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\text{Stab}(g)=C_G(g)
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.5em; vertical-align: -0.5em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1em;"><span class="" style="top: -3.16em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord text"><span class="mord">Stab</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07153em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></div>
<span class="eqnum">(2)</span>
<div class="maths"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">∣</mo><mtext>Orb</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle mathsize="0.9em"><mtext>the number of distinct conjugates of g</mtext></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle mathsize="0.9em"><mtext>the number of permutations with the same cycle structure as </mtext></mstyle><mi>g</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\lvert \text{Orb}(g)\rvert&=\text{\small the number of distinct conjugates of g}\\
&=\text{\small the number of permutations with the same cycle structure as }g
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">∣</span><span class="mord text"><span class="mord">Orb</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)∣</span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5">the number of distinct conjugates of g</span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5">the number of permutations with the same cycle structure as </span></span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></div>
Let <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathnormal">a</span></span></span></span></span> be the product of the lengths of the cycles of <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span></span></span></span></span><br>
Let <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.69444em; vertical-align: 0em;"></span><span class="mord mathnormal">b</span></span></span></span></span> be the number of cycles of equal length.<br>
Let <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.69444em; vertical-align: 0em;"></span><span class="mord mathnormal" style="margin-right: 0.01968em;">l</span></span></span></span></span> be the least common multiple of the lengths of the cycles, <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∣</mo><mi>g</mi><mo stretchy="false">∣</mo><mo>=</mo><mi>l</mi><mo>≤</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\lvert g\rvert=l\leq a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">∣</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.83041em; vertical-align: -0.13597em;"></span><span class="mord mathnormal" style="margin-right: 0.01968em;">l</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.43056em; vertical-align: 0em;"></span><span class="mord mathnormal">a</span></span></span></span></span>.<br><br>
Then the number of permutations with the same cycle structure as <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.19444em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span></span></span></span></span> is
<div class="maths"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mfrac><mrow><mi>n</mi><mo stretchy="false">!</mo></mrow><mrow><mi>a</mi><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">∣</mo><mi>G</mi><mo stretchy="false">∣</mo></mrow><mrow><mi>a</mi><mi>b</mi></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\frac{n!}{ab}=\frac{\lvert G\rvert}{ab}
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 2.413em; vertical-align: -0.9565em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.4565em;"><span class="" style="top: -3.4565em;"><span class="pstrut" style="height: 3.427em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.37144em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">ab</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mclose">!</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.427em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">ab</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">∣</span><span class="mord mathnormal">G</span><span class="mclose">∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.9565em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></div>
Hence
<div class="maths"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">∣</mo><mtext>Orb</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">∣</mo><mi>G</mi><mo stretchy="false">∣</mo></mrow><mrow><mi>a</mi><mi>b</mi></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\lvert \text{Orb}(g)\rvert=\frac{\lvert G\rvert}{ab}
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 2.413em; vertical-align: -0.9565em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.4565em;"><span class="" style="top: -3.4565em;"><span class="pstrut" style="height: 3.427em;"></span><span class="mord"><span class="mopen">∣</span><span class="mord text"><span class="mord">Orb</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.427em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">ab</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">∣</span><span class="mord mathnormal">G</span><span class="mclose">∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.9565em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></div>
<span class="eqnum" style="margin-top: -3em">(3)</span>
By the Orbit-Stabiliser theorem, <span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">∣</mo><mtext>Orb</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo><mo>×</mo><mo stretchy="false">∣</mo><mtext>Stab</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo><mo>=</mo><mo stretchy="false">∣</mo><mi>G</mi><mo stretchy="false">∣</mo></mrow><annotation encoding="application/x-tex">\lvert \text{Orb}(g)\rvert\times\lvert \text{Stab}(g)\rvert=\lvert G\rvert</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">∣</span><span class="mord text"><span class="mord">Orb</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)∣</span><span class="mspace" style="margin-right: 0.222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.222222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">∣</span><span class="mord text"><span class="mord">Stab</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">∣</span><span class="mord mathnormal">G</span><span class="mclose">∣</span></span></span></span></span> so by (3)
<div class="maths"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">∣</mo><mtext>Stab</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">∣</mo><mi>G</mi><mo stretchy="false">∣</mo></mrow><mrow><mo stretchy="false">∣</mo><mtext>Orb</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">∣</mo><mi>G</mi><mo stretchy="false">∣</mo></mrow><mrow><mo stretchy="false">∣</mo><mi>G</mi><mo stretchy="false">∣</mo><mi mathvariant="normal">/</mi><mi>a</mi><mi>b</mi></mrow></mfrac><mo>=</mo><mi>a</mi><mi>b</mi><mo>≥</mo><mo stretchy="false">∣</mo><mi>g</mi><mo stretchy="false">∣</mo><mi>b</mi><mo>≥</mo><mo stretchy="false">∣</mo><mi>g</mi><mo stretchy="false">∣</mo><mo>=</mo><mi>l</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\lvert \text{Stab}(g)\rvert=\frac{\lvert G\rvert}{\lvert \text{Orb}(g)\rvert}=\frac{\lvert G\rvert}{\lvert G\rvert/ab}=ab\geq\lvert g\rvert b\geq\lvert g\rvert=l
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 2.663em; vertical-align: -1.0815em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.5815em;"><span class="" style="top: -3.5815em;"><span class="pstrut" style="height: 3.427em;"></span><span class="mord"><span class="mopen">∣</span><span class="mord text"><span class="mord">Stab</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.427em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">∣</span><span class="mord text"><span class="mord">Orb</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)∣</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">∣</span><span class="mord mathnormal">G</span><span class="mclose">∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.936em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.427em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">∣</span><span class="mord mathnormal">G</span><span class="mclose">∣</span><span class="mord">/</span><span class="mord mathnormal">ab</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">∣</span><span class="mord mathnormal">G</span><span class="mclose">∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.936em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathnormal">ab</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">∣</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">∣</span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">∣</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathnormal" style="margin-right: 0.01968em;">l</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.0815em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></div>
and it follows that
<div class="maths"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">∣</mo><mtext>Stab</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo><mo>=</mo><mo stretchy="false">∣</mo><mi>g</mi><mo stretchy="false">∣</mo>  <mo>⟺</mo>  <mi>a</mi><mo>=</mo><mi>l</mi><mtext> and </mtext><mi>b</mi><mo>=</mo><mn>1</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\lvert \text{Stab}(g)\rvert=\lvert g \rvert\iff a = l \text{ and } b = 1
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.5em; vertical-align: -0.5em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1em;"><span class="" style="top: -3.16em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mopen">∣</span><span class="mord text"><span class="mord">Stab</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mopen">∣</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathnormal" style="margin-right: 0.01968em;">l</span><span class="mord text"><span class="mord"> and </span></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></div>
<span class="eqnum">(4)</span>
But
<div class="maths"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>a</mi><mo>=</mo><mi>l</mi>  <mo>⟺</mo>  <mi>g</mi><mtext> has coprime cycle lengths</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>b</mi><mo>=</mo><mn>1</mn>  <mo>⟺</mo>  <mi>g</mi><mtext> has unequal cycle lengths</mtext></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
a=l\iff g\text{ has coprime cycle lengths}\\
b=1 \iff g \text{ has unequal cycle lengths}
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 3em; vertical-align: -1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.75em;"><span class="" style="top: -3.91em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathnormal" style="margin-right: 0.01968em;">l</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mord text"><span class="mord"> has coprime cycle lengths</span></span></span></span><span class="" style="top: -2.41em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mord text"><span class="mord"> has unequal cycle lengths</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.25em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span></div>
<span class="eqnum" style="margin-top: -4.5em">(5)</span>
<span class="eqnum">(6)</span>
The result now follows from (1), (2), (4), (5) and (6).
<div class="maths"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>g</mi><mo stretchy="false">⟩</mo>  <mo>⟺</mo>  <mo stretchy="false">∣</mo><mtext>Stab</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo><mo>=</mo><mo stretchy="false">∣</mo><mi>g</mi><mo stretchy="false">∣</mo>  <mo>⟺</mo>  <mtext>the cycles of g are of unequal coprime lengths</mtext></mrow><annotation encoding="application/x-tex">
C_G(g)=\langle g\rangle\iff\lvert \text{Stab}(g)\rvert=\lvert g \rvert\iff\text{the cycles of g are of unequal coprime lengths}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.328331em;"><span class="" style="top: -2.55em; margin-left: -0.07153em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">⟨</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">⟩</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">∣</span><span class="mord text"><span class="mord">Stab</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">)∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">∣</span><span class="mord mathnormal" style="margin-right: 0.03588em;">g</span><span class="mclose">∣</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right: 0.277778em;"></span><span class="mspace" style="margin-right: 0.277778em;"></span></span><span class="base"><span class="strut" style="height: 0.88888em; vertical-align: -0.19444em;"></span><span class="mord text"><span class="mord">the cycles of g are of unequal coprime lengths</span></span></span></span></span></span></div>
Cycles of coprime length will have unequal lengths unless they are 1-cycles. Hence we may replace <em>unequal</em> by <em>at most one 1-cycle</em>.
<span class="halmos"><span class="maths"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">■</mi></mrow><annotation encoding="application/x-tex">\blacksquare</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.675em; vertical-align: 0em;"></span><span class="mord amsrm">■</span></span></span></span></span></span><br><br>
<p> </p>
</body></html>