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<title>GAP (TwistedConjugacy) - Chapter 8: Group homomorphisms</title>
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<p id="mathjaxlink" class="pcenter"><a href="chap8_mj.html">[MathJax on]</a></p>
<p><a id="X83702FC27B3C3098" name="X83702FC27B3C3098"></a></p>
<div class="ChapSects"><a href="chap8.html#X83702FC27B3C3098">8 <span class="Heading">Group homomorphisms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X80DDEC8C82E2A4F1">8.1 <span class="Heading">Representatives of homomorphisms between groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X78ADEE0C83819159">8.1-1 RepresentativesAutomorphismClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7A7935B286050886">8.1-2 RepresentativesEndomorphismClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X81E5CF92816BF199">8.1-3 RepresentativesHomomorphismClasses</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X8164A34A86155DFB">8.2 <span class="Heading">Coincidence and fixed point groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X799546928394FF8B">8.2-1 FixedPointGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X780DF6247E3E9190">8.2-2 CoincidenceGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X8084A06782AE362E">8.3 <span class="Heading">Induced and restricted group homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7F6D0625837B7B94">8.3-1 InducedHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7DBA352982923900">8.3-2 RestrictedHomomorphism</a></span>
</div></div>
</div>

<h3>8 <span class="Heading">Group homomorphisms</span></h3>

<p><a id="X80DDEC8C82E2A4F1" name="X80DDEC8C82E2A4F1"></a></p>

<h4>8.1 <span class="Heading">Representatives of homomorphisms between groups</span></h4>

<p>Please note that the functions below are only implemented for finite groups.</p>

<p><a id="X78ADEE0C83819159" name="X78ADEE0C83819159"></a></p>

<h5>8.1-1 RepresentativesAutomorphismClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesAutomorphismClasses</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of the automorphisms of <var class="Arg">G</var> up to composition with inner automorphisms.</p>

<p><a id="X7A7935B286050886" name="X7A7935B286050886"></a></p>

<h5>8.1-2 RepresentativesEndomorphismClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesEndomorphismClasses</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of the endomorphisms of <var class="Arg">G</var> up to composition with inner automorphisms.</p>

<p>This does the same as calling <code class="code">AllHomomorphismClasses(<var class="Arg">G</var>,<var class="Arg">G</var>)</code>, but should be faster for abelian and non-2-generated groups. For 2-generated groups, this function behaves nearly identical to <code class="func">AllHomomorphismClasses</code> (<a href="/home/runner/gap/doc/ref/chap40.html#X7D0C3D5E864CE954"><span class="RefLink">Ref 40.9-2</span></a>).</p>

<p><a id="X81E5CF92816BF199" name="X81E5CF92816BF199"></a></p>

<h5>8.1-3 RepresentativesHomomorphismClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesHomomorphismClasses</code>( <var class="Arg">H</var>, <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of the homomorphisms from <var class="Arg">H</var> to <var class="Arg">G</var>, up to composition with inner automorphisms of <var class="Arg">G</var>.</p>

<p>This does the same as calling <code class="code">AllHomomorphismClasses(<var class="Arg">H</var>,<var class="Arg">G</var>)</code>, but should be faster for abelian and non-2-generated groups. For 2-generated groups, this function behaves nearly identical to <code class="func">AllHomomorphismClasses</code> (<a href="/home/runner/gap/doc/ref/chap40.html#X7D0C3D5E864CE954"><span class="RefLink">Ref 40.9-2</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SymmetricGroup( 6 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Auts := RepresentativesAutomorphismClasses( G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( Auts );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( Auts, IsGroupHomomorphism and IsEndoMapping and IsBijective );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Ends := RepresentativesEndomorphismClasses( G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( Ends );</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( Ends, IsGroupHomomorphism and IsEndoMapping );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">H := SymmetricGroup( 5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homs := RepresentativesHomomorphismClasses( H, G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( Homs );</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( Homs, IsGroupHomomorphism );</span>
true
</pre></div>

<p><a id="X8164A34A86155DFB" name="X8164A34A86155DFB"></a></p>

<h4>8.2 <span class="Heading">Coincidence and fixed point groups</span></h4>

<p><a id="X799546928394FF8B" name="X799546928394FF8B"></a></p>

<h5>8.2-1 FixedPointGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FixedPointGroup</code>( <var class="Arg">endo</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the subgroup of <code class="code">Source(<var class="Arg">endo</var>)</code> consisting of the elements fixed under the endomorphism <var class="Arg">endo</var>.</p>

<p><a id="X780DF6247E3E9190" name="X780DF6247E3E9190"></a></p>

<h5>8.2-2 CoincidenceGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoincidenceGroup</code>( <var class="Arg">hom1</var>, <var class="Arg">hom2</var>[, <var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the subgroup of <code class="code">Source(<var class="Arg">hom1</var>)</code> consisting of the elements <code class="code">h</code> for which <code class="code">h^<var class="Arg">hom1</var></code> = <code class="code">h^<var class="Arg">hom2</var></code> = ...</p>

<p>For infinite non-abelian groups, this function relies on a mixture of the algorithms described in <a href="chapBib.html#biBroma16-a">[Rom16, Thm. 2]</a>, <a href="chapBib.html#biBbkl20-a">[BKL+20, Sec. 5.4]</a> and <a href="chapBib.html#biBroma21-a">[Rom21, Sec. 7]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := GroupHomomorphismByImages( G, G, [ (1,2,5,6,4), (1,2)(3,6)(4,5) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ (2,3,4,5,6), (1,2) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( FixedPointGroup( phi ) );</span>
[ (), (1,2,3,6,5), (1,3,5,2,6), (1,5,6,3,2), (1,6,2,5,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">psi := GroupHomomorphismByImages( H, G, [ (1,2,3,4,5), (1,2) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ (), (1,2) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">khi := GroupHomomorphismByImages( H, G, [ (1,2,3,4,5), (1,2) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ (), (1,2)(3,4) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CoincidenceGroup( psi, khi ) = AlternatingGroup( 5 );</span>
true
</pre></div>

<p><a id="X8084A06782AE362E" name="X8084A06782AE362E"></a></p>

<h4>8.3 <span class="Heading">Induced and restricted group homomorphisms</span></h4>

<p><a id="X7F6D0625837B7B94" name="X7F6D0625837B7B94"></a></p>

<h5>8.3-1 InducedHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InducedHomomorphism</code>( <var class="Arg">epi1</var>, <var class="Arg">epi2</var>, <var class="Arg">hom</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the homomorphism induced by <var class="Arg">hom</var> between the images of <var class="Arg">epi1</var> and <var class="Arg">epi2</var>.</p>

<p>Let <var class="Arg">hom</var> be a group homomorphism from a group <code class="code">H</code> to a group <code class="code">G</code>, let <var class="Arg">epi1</var> be an epimorphism from <code class="code">H</code> to a group <code class="code">Q</code> and let <var class="Arg">epi2</var> be an epimorphism from <code class="code">G</code> to a group <code class="code">P</code> such that the kernel of <var class="Arg">epi1</var> is mapped into the kernel of <var class="Arg">epi2</var> by <var class="Arg">hom</var>. This command returns the homomorphism from <code class="code">Q</code> to <code class="code">P</code> that maps <code class="code">h^<var class="Arg">epi1</var></code> to <code class="code">(h^<var class="Arg">hom</var>)^<var class="Arg">epi2</var></code>, for any element <code class="code">h</code> of <code class="code">H</code>. This function generalises <code class="func">InducedAutomorphism</code> (<a href="/home/runner/gap/doc/ref/chap40.html#X7FC9B6EA7CAADC0A"><span class="RefLink">ref 40.7-7</span></a>) to homomorphisms.</p>

<p><a id="X7DBA352982923900" name="X7DBA352982923900"></a></p>

<h5>8.3-2 RestrictedHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RestrictedHomomorphism</code>( <var class="Arg">hom</var>, <var class="Arg">N</var>, <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the homomorphism <var class="Arg">hom</var>, but restricted as a map from <var class="Arg">N</var> to <var class="Arg">M</var>.</p>

<p>Let <var class="Arg">hom</var> be a group homomorphism from a group <code class="code">H</code> to a group <code class="code">G</code>, and let <var class="Arg">N</var> be subgroup of <code class="code">H</code> such that its image under <var class="Arg">hom</var> is a subgroup of <var class="Arg">M</var>. This command returns the homomorphism from <var class="Arg">N</var> to <var class="Arg">M</var> that maps <code class="code">n</code> to <code class="code">n^<var class="Arg">hom</var></code> for any element <code class="code">n</code> of <var class="Arg">N</var>. No checks are made to verify that <var class="Arg">hom</var> maps <var class="Arg">N</var> into <var class="Arg">M</var>. This function is similar to <code class="func">RestrictedMapping</code> (<a href="/home/runner/gap/doc/ref/chap32.html#X800014D683A81009"><span class="RefLink">ref 32.2-13</span></a>), but its range is explicitly set to <var class="Arg">M</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := PcGroupCode( 1018013, 28 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := GroupHomomorphismByImages( G, G, [ G.1, G.3 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ G.1*G.2*G.3^2, G.3^4 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := DerivedSubgroup( G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p := NaturalHomomorphismByNormalSubgroup( G, N );</span>
[ f1, f2, f3 ] -> [ f1, f2, <identity> of ... ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ind := InducedHomomorphism( p, p, phi );</span>
[ f1 ] -> [ f1*f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Source( ind ) = Range( p ) and Range( ind ) = Range( p );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">res := RestrictedHomomorphism( phi, N, N );</span>
[ f3 ] -> [ f3^4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Source( res ) = N and Range( res ) = N;</span>
true
</pre></div>


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