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<p><a id="X78F9595B78DAC70D" name="X78F9595B78DAC70D"></a></p>
<div class="ChapSects"><a href="chap4_mj.html#X78F9595B78DAC70D">4 <span class="Heading">Twisted conjugacy classes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7CACD3337A7C90F0">4.1 <span class="Heading">Creating a twisted conjugacy class</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79690F4D7F2660B3">4.1-1 TwistedConjugacyClass</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7FA74F8E7BB7915D">4.2 <span class="Heading">Operations on twisted conjugacy classes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X865507568182424E">4.2-1 <span class="Heading">Representative</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B9DB15D80CE28B4">4.2-2 <span class="Heading">ActingDomain</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X86153CB087394DC1">4.2-3 <span class="Heading">FunctionAction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X87BDB89B7AAFE8AD">4.2-4 <span class="Heading">\in</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X858ADA3B7A684421">4.2-5 <span class="Heading">Size</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X867840C67C990840">4.2-6 <span class="Heading">StabiliserOfExternalSet</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7EBA57FC7CCF8449">4.2-7 <span class="Heading">List</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79730D657AB219DB">4.2-8 <span class="Heading">Random</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X806A4814806A4814">4.2-9 <span class="Heading">\=</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X8238998382FE372A">4.3 <span class="Heading">Calculating all twisted conjugacy classes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X797192EA7D30C78F">4.3-1 TwistedConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X862C49C0834E01D7">4.3-2 RepresentativesTwistedConjugacyClasses</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">Twisted conjugacy classes</span></h3>

<p>The orbits of the <span class="SimpleMath">\((\varphi,\psi)\)</span>-twisted conjugacy action are called the <em><span class="SimpleMath">\((\varphi,\psi)\)</span>-twisted conjugacy classes</em> or the <em>Reidemeister classes of <span class="SimpleMath">\((\varphi,\psi)\)</span></em>. We denote the twisted conjugacy class of <span class="SimpleMath">\(g \in G\)</span> by <span class="SimpleMath">\([g]_{\varphi,\psi}\)</span>.</p>

<p><a id="X7CACD3337A7C90F0" name="X7CACD3337A7C90F0"></a></p>

<h4>4.1 <span class="Heading">Creating a twisted conjugacy class</span></h4>

<p><a id="X79690F4D7F2660B3" name="X79690F4D7F2660B3"></a></p>

<h5>4.1-1 TwistedConjugacyClass</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwistedConjugacyClass</code>( <var class="Arg">hom1</var>[, <var class="Arg">hom2</var>], <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReidemeisterClass</code>( <var class="Arg">hom1</var>[, <var class="Arg">hom2</var>], <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the <code class="code">(<var class="Arg">hom1</var>,<var class="Arg">hom2</var>)</code>-twisted conjugacy class of <var class="Arg">g</var>.</p>

<p><a id="X7FA74F8E7BB7915D" name="X7FA74F8E7BB7915D"></a></p>

<h4>4.2 <span class="Heading">Operations on twisted conjugacy classes</span></h4>

<p><a id="X865507568182424E" name="X865507568182424E"></a></p>

<h5>4.2-1 <span class="Heading">Representative</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Representative</code>( <var class="Arg">tcc</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the group element that was used to construct <var class="Arg">tcc</var>.</p>

<p><a id="X7B9DB15D80CE28B4" name="X7B9DB15D80CE28B4"></a></p>

<h5>4.2-2 <span class="Heading">ActingDomain</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ActingDomain</code>( <var class="Arg">tcc</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the group whose twisted conjugacy action <var class="Arg">tcc</var> is an orbit of.</p>

<p><a id="X86153CB087394DC1" name="X86153CB087394DC1"></a></p>

<h5>4.2-3 <span class="Heading">FunctionAction</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FunctionAction</code>( <var class="Arg">tcc</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the twisted conjugacy action that <var class="Arg">tcc</var> is an orbit of.</p>

<p><a id="X87BDB89B7AAFE8AD" name="X87BDB89B7AAFE8AD"></a></p>

<h5>4.2-4 <span class="Heading">\in</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \in</code>( <var class="Arg">g</var>, <var class="Arg">tcc</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">g</var> is an element of <var class="Arg">tcc</var>, otherwise <code class="keyw">false</code>.</p>

<p><a id="X858ADA3B7A684421" name="X858ADA3B7A684421"></a></p>

<h5>4.2-5 <span class="Heading">Size</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Size</code>( <var class="Arg">tcc</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the number of elements in <var class="Arg">tcc</var>.</p>

<p>This is calculated using the orbit-stabiliser theorem.</p>

<p><a id="X867840C67C990840" name="X867840C67C990840"></a></p>

<h5>4.2-6 <span class="Heading">StabiliserOfExternalSet</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StabiliserOfExternalSet</code>( <var class="Arg">tcc</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the stabiliser of <code class="code">Representative(<var class="Arg">tcc</var>)</code> under the action <code class="code">FunctionAction(<var class="Arg">tcc</var>)</code>.</p>

<p><a id="X7EBA57FC7CCF8449" name="X7EBA57FC7CCF8449"></a></p>

<h5>4.2-7 <span class="Heading">List</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ List</code>( <var class="Arg">tcc</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list containing the elements of <var class="Arg">tcc</var>.</p>

<p>If <var class="Arg">tcc</var> is infinite, this will run forever. It is recommended to first test the finiteness of <var class="Arg">tcc</var> using <code class="func">Size</code> (<a href="chap4_mj.html#X858ADA3B7A684421"><span class="RefLink">4.2-5</span></a>).</p>

<p><a id="X79730D657AB219DB" name="X79730D657AB219DB"></a></p>

<h5>4.2-8 <span class="Heading">Random</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Random</code>( <var class="Arg">tcc</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a random element in <var class="Arg">tcc</var>.</p>

<p><a id="X806A4814806A4814" name="X806A4814806A4814"></a></p>

<h5>4.2-9 <span class="Heading">\=</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \=</code>( <var class="Arg">tcc1</var>, <var class="Arg">tcc2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">tcc1</var> is equal to <var class="Arg">tcc2</var>, otherwise <code class="keyw">false</code>.</p>

<p><a id="X8238998382FE372A" name="X8238998382FE372A"></a></p>

<h4>4.3 <span class="Heading">Calculating all twisted conjugacy classes</span></h4>

<p><a id="X797192EA7D30C78F" name="X797192EA7D30C78F"></a></p>

<h5>4.3-1 TwistedConjugacyClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwistedConjugacyClasses</code>( <var class="Arg">hom1</var>[, <var class="Arg">hom2</var>][, <var class="Arg">N</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReidemeisterClasses</code>( <var class="Arg">hom1</var>[, <var class="Arg">hom2</var>][, <var class="Arg">N</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list containing the (<var class="Arg">hom1</var>, <var class="Arg">hom2</var>)-twisted conjugacy classes if there are finitely many, or <code class="keyw">fail</code> otherwise.</p>

<p>If the argument <var class="Arg">N</var> is provided, it must be a normal subgroup of <code class="code">Range(<var class="Arg">hom1</var>)</code>; the function will then only return the Reidemeister classes that intersect <var class="Arg">N</var> non-trivially. It is guaranteed that the Reidemeister class of the identity is in the first position, and that the representatives of the classes belong to <var class="Arg">N</var> if this argument is provided.</p>

<p>If <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(H\)</span> are finite, this function relies on an orbit-stabiliser algorithm. Otherwise, it relies on the algorithms in <a href="chapBib_mj.html#biBdt21-a">[DT21]</a> and <a href="chapBib_mj.html#biBtert25-a">[Ter25]</a>.</p>

<p><a id="X862C49C0834E01D7" name="X862C49C0834E01D7"></a></p>

<h5>4.3-2 RepresentativesTwistedConjugacyClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesTwistedConjugacyClasses</code>( <var class="Arg">hom1</var>[, <var class="Arg">hom2</var>][, <var class="Arg">N</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesReidemeisterClasses</code>( <var class="Arg">hom1</var>[, <var class="Arg">hom2</var>][, <var class="Arg">N</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list containing representatives of the (<var class="Arg">hom1</var>, <var class="Arg">hom2</var>)-twisted conjugacy classes if there are finitely many, or <code class="keyw">fail</code> otherwise.</p>

<p>If the argument <var class="Arg">N</var> is provided, it must be a normal subgroup of <code class="code">Range(<var class="Arg">hom1</var>)</code>; the function will then only return the representatives of the twisted conjugacy classes that intersect <var class="Arg">N</var> non-trivially. It is guaranteed that the identity is in the first position, and that all elements belong to <var class="Arg">N</var> if this argument is provided.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tcc := TwistedConjugacyClass( phi, psi, g1 );</span>
(4,6,5)^G
<span class="GAPprompt">gap></span> <span class="GAPinput">Representative( tcc );</span>
(4,6,5)
<span class="GAPprompt">gap></span> <span class="GAPinput">ActingDomain( tcc ) = H;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">FunctionAction( tcc )( g1, h );</span>
(1,6,4,2)(3,5)
<span class="GAPprompt">gap></span> <span class="GAPinput">List( tcc );</span>
[ (4,6,5), (1,6,4,2)(3,5) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( tcc );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">StabiliserOfExternalSet( tcc );</span>
Group([ (1,2,3,4,5), (1,3,4,5,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">TwistedConjugacyClasses( phi, psi ){[1..7]};</span>
[ ()^G, (4,5,6)^G, (4,6,5)^G, (3,4)(5,6)^G, (3,4,5)^G, (3,4,6)^G, (3,5,4)^G ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativesTwistedConjugacyClasses( phi, psi ){[1..7]};</span>
[ (), (4,5,6), (4,6,5), (3,4)(5,6), (3,4,5), (3,4,6), (3,5,4) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">NrTwistedConjugacyClasses( phi, psi );</span>
184
</pre></div>


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