<p>Let <span class="Math">G</span> and <span class="Math">H</span> be groups and let <span class="Math">\varphi</span> and <span class="Math">\psi</span> be group homomorphisms from <span class="Math">H</span> to <span class="Math">G</span>. The pair <span class="Math">(\varphi,\psi)</span> induces a (right) group action of <span class="Math">H</span> on <span class="Math">G</span> given by the map</p>
<p class="pcenter">G \times H \to G
\colon (g,h) \mapsto \varphi(h)^{-1} g\,\psi(h).</p>
<p>This group action is called <em><span class="Math">(\varphi,\psi)</span>-twisted conjugation</em>.</p>
<p>If <span class="Math">G = H</span>, <span class="Math">\varphi</span> is an endomorphism of <span class="Math">G</span> and <span class="Math">\psi = \operatorname{id}_G</span>, then the action is usually called <em><span class="Math">\varphi</span>-twisted conjugation</em>. In general, for the <strong class="pkg">TwistedConjugacy</strong> package, many functions will take two homomorphisms <var class="Arg">hom1</var> and <var class="Arg">hom2</var> as arguments. However, if <var class="Arg">hom1</var> is an endomorphism, <var class="Arg">hom2</var> can be omitted, in which case it is automatically taken to be the identity map.</p>
<p>Similarly, some functions will take two elements <var class="Arg">g1</var> and <var class="Arg">g2</var> as arguments. If <var class="Arg">g2</var> is omitted, it is automatically taken to be the identity element.</p>
<p>Given groups <span class="Math">G</span> and <span class="Math">H</span>, group homomorphisms <span class="Math">\varphi</span> and <span class="Math">\psi</span> from <span class="Math">H</span> to <span class="Math">G</span> and elements <span class="Math">g_1, g_2 \in G</span>, the <em>twisted conjugacy problem</em> is the decision problem that asks whether <span class="Math">g_1</span> and <span class="Math">g_2</span> are <span class="Math">(\varphi,\psi)</span>-twisted conjugate. The <em>twisted conjugacy search problem</em> is the problem of determining an explicit <span class="Math">h</span> such that <span class="Math">\varphi(h)^{-1}g_1\psi(h) = g_2</span> (under the assumption that such <span class="Math">h</span> exists).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativeTwistedConjugation</code>( <var class="Arg">hom1</var>[, <var class="Arg">hom2</var>], <var class="Arg">g1</var>[, <var class="Arg">g2</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an element that maps <var class="Arg">g1</var> to <var class="Arg">g2</var> under the <codeclass="code">(<var class="Arg">hom1</var>,<var class="Arg">hom2</var>)</code>-twisted conjugacy action, or <code class="keyw">fail</code> if no such element exists.</p>
<p>If the source group is finite, this function relies on orbit-stabiliser algorithms provided by <strong class="pkg">GAP</strong>. Otherwise, it relies on a mixture of the algorithms described in <a href="chapBib.html#biBroma16-a">[Rom16, Thm. 3]</a>, <a href="chapBib.html#biBbkl20-a">[BKL+20, Sec. 5.4]</a>, <a href="chapBib.html#biBroma21-a">[Rom21, Sec. 7]</a> and <a href="chapBib.html#biBdt21-a">[DT21]</a>.</p>
<p>Let <span class="Math">H</span> and <span class="Math">G_1, \ldots, G_n</span> be groups. For each <span class="Math">i \in \{1,\ldots,n\}</span>, let <span class="Math">g_i,g_i' \in G_i and let \varphi_i,\psi_i\colon H \to G_i be group homomorphisms. The multiple twisted conjugacy problem is the decision problem that asks whether there exists some h \in H such that \varphi_i(h)^{-1}g_i\psi_i(h) = g_i'</span> for all <span class="Math">i \in \{1,\ldots,n\}</span>. The <em>multiple twisted conjugacy search problem</em> is the problem of determining an explicit <span class="Math">h</span> such that <span class="Math">\varphi_i(h)^{-1}g_i\psi_i(h) = g_i' for all i \in \{1,\ldots,n\} (under the assumption that such h exists).
<p><code class="func">IsTwistedConjugate</code> (<a href="chap3.html#X809D34107CFE8082"><span class="RefLink">3.2-1</span></a>) and <code class="func">RepresentativeTwistedConjugation</code> (<a href="chap3.html#X8493E3818276A562"><span class="RefLink">3.2-2</span></a>) can take lists instead of their usual arguments to solve these problems.</p>
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