<p>The orbits of the <span class="Math">(\varphi,\psi)</span>-twisted conjugacy action are called the <em><span class="Math">(\varphi,\psi)</span>-twisted conjugacy classes</em> or the <em>Reidemeister classes of <span class="Math">(\varphi,\psi)</span></em>. We denote the twisted conjugacy class of <span class="Math">g \in G</span> by <span class="Math">[g]_{\varphi,\psi}</span>.</p>
<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>
<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>
<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.html#X858ADA3B7A684421"><span class="RefLink">4.2-5</span></a>).</p>
<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>
<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="Math">G</span> and <span class="Math">H</span> are finite, this function relies on an orbit-stabiliser algorithm. Otherwise, it relies on the algorithms in <a href="chapBib.html#biBdt21-a">[DT21]</a> and <a href="chapBib.html#biBtert25-a">[Ter25]</a>.</p>
<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>
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