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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap39.html">[Previous Chapter]</a> <a href="chap41.html">[Next Chapter]</a> </div>
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<div class="ChapSects"><a href="chap40.html#X83702FC27B3C3098">40 <span class="Heading">Group Homomorphisms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X81A7BB0F7D81B247">40.1 <span class="Heading">Creating Group Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7F348F497C813BE0">40.1-1 GroupHomomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7AB15AF5830F2A6B">40.1-2 GroupHomomorphismByImagesNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7A59F2C47BD41DC8">40.1-3 GroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7BC6C20E7CEDBFC5">40.1-4 <span class="Heading">GroupHomomorphismByFunction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X785AB6057F736344">40.1-5 AsGroupGeneralMappingByImages</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X794043AC7E4FAF9E">40.2 <span class="Heading">Operations for Group Homomorphisms</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X7A121B9E7F78138A">40.3 <span class="Heading">Efficiency of Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X84CFBB577BAFFD4D">40.3-1 <span class="Heading">Mappings given on generators</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X86C2BE2481FDC8EE">40.3-2 <span class="Heading">Action homomorphisms</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X802C5A887D8A7CC4">40.3-3 <span class="Heading">Mappings given by functions</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X87497C207B7D7511">40.3-4 <span class="Heading">Other operations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X80B8ABEC7CC20DFB">40.3-5 ImagesSmallestGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X7BA90DA481A1C6D6">40.4 <span class="Heading">Homomorphism for very large groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X7FFD731684606BC6">40.5 <span class="Heading">Nice Monomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X78849F81804C44B3">40.5-1 IsHandledByNiceMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7965086E82ABCF41">40.5-2 NiceMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7B47BE0983E93A83">40.5-3 NiceObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X8652149F7F291EE3">40.5-4 IsCanonicalNiceMonomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X783030917CB43959">40.6 <span class="Heading">Group Automorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7E52E99487562F3A">40.6-1 ConjugatorIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X79ED68CF8139F08A">40.6-2 ConjugatorAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7E937A947856D9DA">40.6-3 InnerAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7F31FECC7A3D4A8A">40.6-4 IsConjugatorIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X78FE7E307E86525A">40.6-5 ConjugatorOfConjugatorIsomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X79640F3682BDBFC1">40.7 <span class="Heading">Groups of Automorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X87677B0787B4461A">40.7-1 AutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7FC631B786C1DC8B">40.7-2 IsGroupOfAutomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7B767B9D827EB0FC">40.7-3 AutomorphismDomain</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7F87D5957D9B991E">40.7-4 IsAutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X8476738A7BF9BADA">40.7-5 InnerAutomorphismsAutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7FC9B6EA7CAADC0A">40.7-6 InducedAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X7A8E961C7F1A57B3">40.8 <span class="Heading">Calculating with Group Automorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X85691E8386107403">40.8-1 AssignNiceMonomorphismAutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7C9FB0A57BFF6CC0">40.8-2 NiceMonomorphismAutomGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X81B79CC27F47D429">40.9 <span class="Heading">Searching for Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7B536A32827788C6">40.9-1 IsomorphismGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7D0C3D5E864CE954">40.9-2 AllHomomorphismClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X791D12B7845610CE">40.9-3 AllHomomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X790C261184EEAB94">40.9-4 GQuotients</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X83B417BE7C508DC4">40.9-5 IsomorphicSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7AABA9A27E30BF2B">40.9-6 MorClassLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X81FC3CEF85CED3DC">40.10 <span class="Heading">Representations for Group Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X82B77A5F7F9EDBC7">40.10-1 IsGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X863805187A24B5E3">40.10-2 MappingGeneratorsImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7DFBBAB18126B4D9">40.10-3 IsGroupGeneralMappingByAsGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X78707DF57C3927EB">40.10-4 IsPreimagesByAsGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X83E10338798F552B">40.10-5 IsPermGroupGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X83DADD9F7CAD829B">40.10-6 IsToPermGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X798E72E77EC85D4A">40.10-7 IsGroupGeneralMappingByPcgs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X86FF63B784FB8F85">40.10-8 IsPcGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X79A853B579B250C0">40.10-9 IsToPcGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7BE2A2EB80DC5CFF">40.10-10 IsFromFpGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X81090C207F4F6423">40.10-11 IsFromFpGroupStdGensGeneralMappingByImages</a></span>
</div></div>
</div>
<h3>40 <span class="Heading">Group Homomorphisms</span></h3>
<p>A group homomorphism is a mapping from one group to another that respects multiplication and inverses. They are implemented as a special class of mappings, so in particular all operations for mappings, such as <code class="func">Image</code> (<a href="chap32.html#X87F4D35A826599C6"><span class="RefLink">32.4-6</span></a>), <code class="func">PreImage</code> (<a href="chap32.html#X836FAEAC78B55BF4"><span class="RefLink">32.5-6</span></a>), <code class="func">PreImagesRepresentative</code> (<a href="chap32.html#X7AE24A1586B7DE79"><span class="RefLink">32.5-4</span></a>), <code class="func">KernelOfMultiplicativeGeneralMapping</code> (<a href="chap32.html#X81A5A5CF846E5FBF"><span class="RefLink">32.9-5</span></a>), <code class="func">Source</code> (<a href="chap32.html#X7DE8173F80E07AB1"><span class="RefLink">32.3-8</span></a>), <code class="func">Range</code> (<a href="chap32.html#X7B6FD7277CDE9FCB"><span class="RefLink">32.3-7</span></a>), <code class="func">IsInjective</code> (<a href="chap32.html#X7F065FD7822C0A12"><span class="RefLink">32.3-4</span></a>) and <code class="func">IsSurjective</code> (<a href="chap32.html#X784ECE847E005B8F"><span class="RefLink">32.3-5</span></a>) (see chapter <a href="chap32.html#X7C9734B880042C73"><span class="RefLink">32</span></a>, in particular section <a href="chap32.html#X806F892C862F29F9"><span class="RefLink">32.9</span></a>) are applicable to them.</p>
<p>Homomorphisms can be used to transfer calculations into isomorphic groups in another representation, for which better algorithms are available. Section <a href="chap40.html#X7FFD731684606BC6"><span class="RefLink">40.5</span></a> explains a technique how to enforce this automatically.</p>
<p>Homomorphisms are also used to represent group automorphisms, and section <a href="chap40.html#X783030917CB43959"><span class="RefLink">40.6</span></a> explains explains <strong class="pkg">GAP</strong>'s facilities to work with automorphism groups.</p>
<p>Section <a href="chap40.html#X81B79CC27F47D429"><span class="RefLink">40.9</span></a> explains how to make <strong class="pkg">GAP</strong> to search for all homomorphisms between two groups which fulfill certain specifications.</p>
<p><a id="X81A7BB0F7D81B247" name="X81A7BB0F7D81B247"></a></p>
<h4>40.1 <span class="Heading">Creating Group Homomorphisms</span></h4>
<p>The most important way of creating group homomorphisms is to give images for a set of group generators and to extend it to the group generated by them by the homomorphism property.</p>
<p><em>A second</em> way to create homomorphisms is to give functions that compute image and preimage. (A similar case are homomorphisms that are induced by conjugation. Special constructors for such mappings are described in section <a href="chap40.html#X783030917CB43959"><span class="RefLink">40.6</span></a>).</p>
<p><em>The third</em> class are epimorphisms from a group onto its factor group. Such homomorphisms can be constructed by <code class="func">NaturalHomomorphismByNormalSubgroup</code> (<a href="chap39.html#X80FC390C7F38A13F"><span class="RefLink">39.18-1</span></a>).</p>
<p><em>The fourth</em> class is homomorphisms in a permutation group that are induced by an action on a set. Such homomorphisms are described in the context of group actions, see chapter <a href="chap41.html#X87115591851FB7F4"><span class="RefLink">41</span></a> and in particular <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>).</p>
<p><a id="X7F348F497C813BE0" name="X7F348F497C813BE0"></a></p>
<h5>40.1-1 GroupHomomorphismByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupHomomorphismByImages</code>( <var class="Arg">G</var>, <var class="Arg">H</var>[[, <var class="Arg">gens</var>], <var class="Arg">imgs</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">GroupHomomorphismByImages</code> returns the group homomorphism with source <var class="Arg">G</var> and range <var class="Arg">H</var> that is defined by mapping the list <var class="Arg">gens</var> of generators of <var class="Arg">G</var> to the list <var class="Arg">imgs</var> of images in <var class="Arg">H</var>.</p>
<p>If omitted, the arguments <var class="Arg">gens</var> and <var class="Arg">imgs</var> default to the <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>) value of <var class="Arg">G</var> and <var class="Arg">H</var>, respectively. If <var class="Arg">H</var> is not given the mapping is automatically considered as surjective.</p>
<p>If <var class="Arg">gens</var> does not generate <var class="Arg">G</var> or if the mapping of the generators does not extend to a homomorphism (i.e., if mapping the generators describes only a multi-valued mapping) then <code class="keyw">fail</code> is returned.</p>
<p>This test can be quite expensive. If one is certain that the mapping of the generators extends to a homomorphism, one can avoid the checks by calling <code class="func">GroupHomomorphismByImagesNC</code> (<a href="chap40.html#X7AB15AF5830F2A6B"><span class="RefLink">40.1-2</span></a>). (There also is the possibility to construct potentially multi-valued mappings with <code class="func">GroupGeneralMappingByImages</code> (<a href="chap40.html#X7A59F2C47BD41DC8"><span class="RefLink">40.1-3</span></a>) and to test with <code class="func">IsMapping</code> (<a href="chap32.html#X7CC95EB282854385"><span class="RefLink">32.3-3</span></a>) whether they are indeed homomorphisms.)</p>
<p><a id="X7AB15AF5830F2A6B" name="X7AB15AF5830F2A6B"></a></p>
<h5>40.1-2 GroupHomomorphismByImagesNC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupHomomorphismByImagesNC</code>( <var class="Arg">G</var>, <var class="Arg">H</var>[[, <var class="Arg">gens</var>], <var class="Arg">imgs</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">GroupHomomorphismByImagesNC</code> creates a homomorphism as <code class="func">GroupHomomorphismByImages</code> (<a href="chap40.html#X7F348F497C813BE0"><span class="RefLink">40.1-1</span></a>) does, however it does not test whether <var class="Arg">gens</var> generates <var class="Arg">G</var> and that the mapping of <var class="Arg">gens</var> to <var class="Arg">imgs</var> indeed defines a group homomorphism. Because these tests can be expensive it can be substantially faster than <code class="func">GroupHomomorphismByImages</code> (<a href="chap40.html#X7F348F497C813BE0"><span class="RefLink">40.1-1</span></a>). Results are unpredictable if the conditions do not hold.</p>
<p>If omitted, the arguments <var class="Arg">gens</var> and <var class="Arg">imgs</var> default to the <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>) value of <var class="Arg">G</var> and <var class="Arg">H</var>, respectively.</p>
<p>(For creating a possibly multi-valued mapping from <var class="Arg">G</var> to <var class="Arg">H</var> that respects multiplication and inverses, <code class="func">GroupGeneralMappingByImages</code> (<a href="chap40.html#X7A59F2C47BD41DC8"><span class="RefLink">40.1-3</span></a>) can be used.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:=[(1,2,3,4),(1,2)];</span>
[ (1,2,3,4), (1,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group(gens);</span>
Group([ (1,2,3,4), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Group((1,2,3),(1,2));</span>
Group([ (1,2,3), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=GroupHomomorphismByImages(g,h,gens,[(1,2),(1,3)]);</span>
[ (1,2,3,4), (1,2) ] -> [ (1,2), (1,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Image(hom,(1,4));</span>
(2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">map:=GroupHomomorphismByImages(g,h,gens,[(1,2,3),(1,2)]);</span>
fail
</pre></div>
<p><a id="X7A59F2C47BD41DC8" name="X7A59F2C47BD41DC8"></a></p>
<h5>40.1-3 GroupGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupGeneralMappingByImages</code>( <var class="Arg">G</var>, <var class="Arg">H</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupGeneralMappingByImages</code>( <var class="Arg">G</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupGeneralMappingByImagesNC</code>( <var class="Arg">G</var>, <var class="Arg">H</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupGeneralMappingByImagesNC</code>( <var class="Arg">G</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a general mapping defined by extending the mapping from <var class="Arg">gens</var> to <var class="Arg">imgs</var> homomorphically. If the range <var class="Arg">H</var> is not given the mapping will be made automatically surjective. The NC version does not test whether <var class="Arg">gens</var> are contained in <var class="Arg">G</var> or <var class="Arg">imgs</var> are contained in <var class="Arg">H</var>. (<code class="func">GroupHomomorphismByImages</code> (<a href="chap40.html#X7F348F497C813BE0"><span class="RefLink">40.1-1</span></a>) creates a group general mapping by images and tests whether it is in <code class="func">IsMapping</code> (<a href="chap32.html#X7CC95EB282854385"><span class="RefLink">32.3-3</span></a>).)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">map:=GroupGeneralMappingByImages(g,h,gens,[(1,2,3),(1,2)]);</span>
[ (1,2,3,4), (1,2) ] -> [ (1,2,3), (1,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMapping(map);</span>
false
</pre></div>
<p><a id="X7BC6C20E7CEDBFC5" name="X7BC6C20E7CEDBFC5"></a></p>
<h5>40.1-4 <span class="Heading">GroupHomomorphismByFunction</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupHomomorphismByFunction</code>( <var class="Arg">S</var>, <var class="Arg">R</var>, <var class="Arg">fun</var>[, <var class="Arg">invfun</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">‣ GroupHomomorphismByFunction</code>( <var class="Arg">S</var>, <var class="Arg">R</var>, <var class="Arg">fun</var>, <var class="Arg">false</var>, <var class="Arg">prefun</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">GroupHomomorphismByFunction</code> returns a group homomorphism <code class="code">hom</code> with source <var class="Arg">S</var> and range <var class="Arg">R</var>, such that each element <code class="code">s</code> of <var class="Arg">S</var> is mapped to the element <var class="Arg">fun</var><code class="code">( s )</code>, where <var class="Arg">fun</var> is a <strong class="pkg">GAP</strong> function.</p>
<p>If the argument <var class="Arg">invfun</var> is bound then <var class="Arg">hom</var> is a bijection between <var class="Arg">S</var> and <var class="Arg">R</var>, and the preimage of each element <code class="code">r</code> of <var class="Arg">R</var> is given by <var class="Arg">invfun</var><code class="code">( r )</code>, where <var class="Arg">invfun</var> is a <strong class="pkg">GAP</strong> function.</p>
<p>If five arguments are given and the fourth argument is <code class="keyw">false</code> then the <strong class="pkg">GAP</strong> function <var class="Arg">prefun</var> can be used to compute a single preimage also if <code class="code">hom</code> is not bijective.</p>
<p>No test is performed on whether the functions actually give an homomorphism between both groups because this would require testing the full multiplication table.</p>
<p><code class="func">GroupHomomorphismByFunction</code> creates a mapping which lies in <code class="func">IsSPGeneralMapping</code> (<a href="chap32.html#X7D28581F82481163"><span class="RefLink">32.14-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=GroupHomomorphismByFunction(g,h,</span>
<span class="GAPprompt">></span> <span class="GAPinput">function(x) if SignPerm(x)=-1 then return (1,2); else return ();fi;end);</span>
MappingByFunction( Group([ (1,2,3,4), (1,2) ]), Group(
[ (1,2,3), (1,2) ]), function( x ) ... end )
<span class="GAPprompt">gap></span> <span class="GAPinput">ImagesSource(hom);</span>
Group([ (1,2), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Image(hom,(1,2,3,4));</span>
(1,2)
</pre></div>
<p><a id="X785AB6057F736344" name="X785AB6057F736344"></a></p>
<h5>40.1-5 AsGroupGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsGroupGeneralMappingByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">map</var> is a mapping from one group to another this attribute returns a group general mapping that which implements the same abstract mapping. (Some operations can be performed more effective in this representation, see also <code class="func">IsGroupGeneralMappingByAsGroupGeneralMappingByImages</code> (<a href="chap40.html#X7DFBBAB18126B4D9"><span class="RefLink">40.10-3</span></a>).)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsGroupGeneralMappingByImages(hom);</span>
[ (1,2,3,4), (1,2) ] -> [ (1,2), (1,2) ]
</pre></div>
<p><a id="X794043AC7E4FAF9E" name="X794043AC7E4FAF9E"></a></p>
<h4>40.2 <span class="Heading">Operations for Group Homomorphisms</span></h4>
<p>Group homomorphisms are mappings, so all the operations and properties for mappings described in chapter <a href="chap32.html#X7C9734B880042C73"><span class="RefLink">32</span></a> are applicable to them. (However often much better methods, than for general mappings are available.)</p>
<p>Group homomorphisms will map groups to groups by just mapping the set of generators.</p>
<p><code class="func">KernelOfMultiplicativeGeneralMapping</code> (<a href="chap32.html#X81A5A5CF846E5FBF"><span class="RefLink">32.9-5</span></a>) can be used to compute the kernel of a group homomorphism.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=GroupHomomorphismByImages(g,h,gens,[(1,2),(1,3)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Kernel(hom);</span>
Group([ (1,4)(2,3), (1,2)(3,4) ])
</pre></div>
<p>Homomorphisms can map between groups in different representations and are also used to get isomorphic groups in a different representation.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:=[[0,-1],[1,0]];;m2:=[[0,-1],[1,1]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sl2z:=Group(m1,m2);; # SL(2,Integers) as matrix group</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FreeGroup(2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">psl2z:=F/[F.1^2,F.2^3]; #PSL(2,Z) as FP group</span>
<fp group on the generators [ f1, f2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">phom:=GroupHomomorphismByImagesNC(sl2z,psl2z,[m1,m2],</span>
<span class="GAPprompt">></span> <span class="GAPinput">GeneratorsOfGroup(psl2z)); # the non NC-version would be expensive</span>
[ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 0, -1 ], [ 1, 1 ] ] ] -> [ f1, f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Kernel(phom); # the diagonal matrices</span>
Group([ [ [ -1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput">p1:=(1,2)(3,4);;p2:=(2,4,5);;a5:=Group(p1,p2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ahom:=GroupHomomorphismByImages(psl2z,a5,</span>
<span class="GAPprompt">></span> <span class="GAPinput">GeneratorsOfGroup(psl2z),[p1,p2]); # here homomorphism test is cheap.</span>
[ f1, f2 ] -> [ (1,2)(3,4), (2,4,5) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">u:=PreImage(ahom,Group((1,2,3),(1,2)(4,5)));</span>
Group(<fp, no generators known>)
<span class="GAPprompt">gap></span> <span class="GAPinput">Index(psl2z,u);</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">isofp:=IsomorphismFpGroup(u);; Image(isofp);</span>
<fp group of size infinity on the generators [ F1, F2, F3, F4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">RelatorsOfFpGroup(Image(isofp));</span>
[ F1^2, F4^2, F3^3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">up:=PreImage(phom,u);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(GeneratorsOfGroup(up),TraceMat);</span>
[ -2, -2, 0, -4, 1, 0 ]
</pre></div>
<p>For an automorphism <var class="Arg">aut</var>, <code class="func">Inverse</code> (<a href="chap31.html#X78EE524E83624057"><span class="RefLink">31.10-8</span></a>) returns the inverse automorphism <span class="SimpleMath"><var class="Arg">aut</var>^{-1}</span>. However if <var class="Arg">hom</var> is a bijective homomorphism between different groups, or if <var class="Arg">hom</var> is injective and considered to be a bijection to its image, the operation <code class="func">InverseGeneralMapping</code> (<a href="chap32.html#X865FC25A87D36F3D"><span class="RefLink">32.2-3</span></a>) should be used instead. (See <code class="func">Inverse</code> (<a href="chap31.html#X78EE524E83624057"><span class="RefLink">31.10-8</span></a>) for a further discussion of this problem.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">iso:=IsomorphismPcGroup(g);</span>
Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) -> [ f1, f2, f3, f4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Inverse(iso);</span>
#I The mapping must be bijective and have source=range
#I You might want to use `InverseGeneralMapping'
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">InverseGeneralMapping(iso);</span>
[ f1, f2, f3, f4 ] -> Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ])
</pre></div>
<p><a id="X7A121B9E7F78138A" name="X7A121B9E7F78138A"></a></p>
<h4>40.3 <span class="Heading">Efficiency of Homomorphisms</span></h4>
<p><strong class="pkg">GAP</strong> permits to create homomorphisms between arbitrary groups. This section considers the efficiency of the implementation and shows ways how to choose suitable representations. For permutation groups (see <a href="chap43.html#X85ED46007CED6191"><span class="RefLink">43</span></a>) or Pc groups (see <a href="chap46.html#X7EAD57C97EBF7E67"><span class="RefLink">46</span></a>) this is normally nothing to worry about, unless the groups get extremely large. For other groups however certain calculations might be expensive and some precaution might be needed to avoid unnecessarily expensive calculations.</p>
<p>In short, it is always worth to tell a mapping that it is a homomorphism (this can be done by calling <code class="code">SetIsMapping</code>) (or to create it directly with <code class="func">GroupHomomorphismByImagesNC</code> (<a href="chap40.html#X7AB15AF5830F2A6B"><span class="RefLink">40.1-2</span></a>)).</p>
<p>The basic operations required are to compute image and preimage of elements and to test whether a mapping is a homomorphism. Their cost will differ depending on the type of the mapping.</p>
<p><a id="X84CFBB577BAFFD4D" name="X84CFBB577BAFFD4D"></a></p>
<h5>40.3-1 <span class="Heading">Mappings given on generators</span></h5>
<p>See <code class="func">GroupHomomorphismByImages</code> (<a href="chap40.html#X7F348F497C813BE0"><span class="RefLink">40.1-1</span></a>) and <code class="func">GroupGeneralMappingByImages</code> (<a href="chap40.html#X7A59F2C47BD41DC8"><span class="RefLink">40.1-3</span></a>).</p>
<p>Computing images requires to express an element of the source as word in the generators. If it cannot be done effectively (this is determined by <code class="func">KnowsHowToDecompose</code> (<a href="chap39.html#X87D62C2C7C375E2D"><span class="RefLink">39.25-7</span></a>) which returns <code class="keyw">true</code> for example for arbitrary permutation groups, for Pc groups or for finitely presented groups with the images of the free generators) the span of the generators has to be computed elementwise which can be very expensive and memory consuming.</p>
<p>Computing preimages adheres to the same rules with swapped rôles of generators and their images.</p>
<p>The test whether a mapping is a homomorphism requires the computation of a presentation for the source and evaluation of its relators in the images of its generators. For larger groups this can be expensive and <code class="func">GroupHomomorphismByImagesNC</code> (<a href="chap40.html#X7AB15AF5830F2A6B"><span class="RefLink">40.1-2</span></a>) should be used if the mapping is known to be a homomorphism.</p>
<p><a id="X86C2BE2481FDC8EE" name="X86C2BE2481FDC8EE"></a></p>
<h5>40.3-2 <span class="Heading">Action homomorphisms</span></h5>
<p>See <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>).</p>
<p>The calculation of images is determined by the acting function used and –for large domains– is often dominated by the search for the position of an image in a list of the domain elements. This can be improved by sorting this list if an efficient method for <code class="func">\<</code> (<a href="chap31.html#X7EF67D047F03CA6F"><span class="RefLink">31.11-1</span></a>) to compare elements of the domain is available.</p>
<p>Once the images of a generating set are computed, computing preimages (which is done via <code class="func">AsGroupGeneralMappingByImages</code> (<a href="chap40.html#X785AB6057F736344"><span class="RefLink">40.1-5</span></a>)) and computing the kernel behaves the same as for a homomorphism created with <code class="func">GroupHomomorphismByImages</code> (<a href="chap40.html#X7F348F497C813BE0"><span class="RefLink">40.1-1</span></a>) from a permutation group.</p>
<p><strong class="pkg">GAP</strong> will always assume that the acting function provided implements a proper group action and thus that the mapping is indeed a homomorphism.</p>
<p><a id="X802C5A887D8A7CC4" name="X802C5A887D8A7CC4"></a></p>
<h5>40.3-3 <span class="Heading">Mappings given by functions</span></h5>
<p>See <code class="func">GroupHomomorphismByFunction</code> (<a href="chap40.html#X7BC6C20E7CEDBFC5"><span class="RefLink">40.1-4</span></a>).</p>
<p>Computing images is wholly determined by the function that performs the image calculation. If no function to compute preimages is given, computing preimages requires mapping every element of the source to find an element that maps to the requested image. This is time and memory consuming.</p>
<p><a id="X87497C207B7D7511" name="X87497C207B7D7511"></a></p>
<h5>40.3-4 <span class="Heading">Other operations</span></h5>
<p>To compute the kernel of a homomorphism (unless the mapping is known to be injective) requires the capability to compute a presentation of the image and to evaluate the relators of this presentation in preimages of the presentations generators.</p>
<p>The calculation of the <code class="func">Image</code> (<a href="chap32.html#X87F4D35A826599C6"><span class="RefLink">32.4-6</span></a>) (respectively <code class="func">ImagesSource</code> (<a href="chap32.html#X7D23C1CE863DACD8"><span class="RefLink">32.4-1</span></a>)) value requires to map a generating set of the source, testing surjectivity is a comparison for equality with the range.</p>
<p>Testing injectivity is a test for triviality of the kernel.</p>
<p>The comparison of mappings is based on a lexicographic comparison of a sorted element list of the source. For group homomorphisms, this can be simplified, using <code class="func">ImagesSmallestGenerators</code> (<a href="chap40.html#X80B8ABEC7CC20DFB"><span class="RefLink">40.3-5</span></a>)</p>
<p><a id="X80B8ABEC7CC20DFB" name="X80B8ABEC7CC20DFB"></a></p>
<h5>40.3-5 ImagesSmallestGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImagesSmallestGenerators</code>( <var class="Arg">map</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the list of images of <code class="code">GeneratorsSmallest(Source(<var class="Arg">map</var>))</code>. This list can be used to compare group homomorphisms. (The standard comparison is to compare the image lists on the set of elements of the source. If however x and y have the same images under a and b, certainly all their products have. Therefore it is sufficient to test this on the images of the smallest generators.)</p>
<p><a id="X7BA90DA481A1C6D6" name="X7BA90DA481A1C6D6"></a></p>
<h4>40.4 <span class="Heading">Homomorphism for very large groups</span></h4>
<p>Some homomorphisms (notably particular actions) transfer known information about the source group (such as a stabilizer chain) to the image group if this is substantially cheaper than to compute the information in the image group anew. In most cases this is no problem and in fact speeds up further calculations notably.</p>
<p>For a huge source group, however this can be time consuming or take a large amount of extra memory for storage. In this case it can be helpful to avoid as much automatism as possible.</p>
<p>The following list of tricks might be useful in such a case. (However you will lose much automatic deduction. So please restrict the use of these to cases where the standard approach does not work.)</p>
<ul>
<li><p>Compute only images (or the <code class="func">PreImagesRepresentative</code> (<a href="chap32.html#X7AE24A1586B7DE79"><span class="RefLink">32.5-4</span></a>)) of group elements. Do not compute the images of (sub)groups or the full preimage of a subgroup.</p>
</li>
<li><p>Create action homomorphisms as "surjective" (see <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>)), otherwise the range is set to be the full symmetric group. However do not compute <code class="func">Range</code> (<a href="chap32.html#X7B6FD7277CDE9FCB"><span class="RefLink">32.3-7</span></a>) or <code class="func">Image</code> (<a href="chap32.html#X87F4D35A826599C6"><span class="RefLink">32.4-6</span></a>) values, but only the images of a generator set.</p>
</li>
<li><p>If you suspect an action homomorphism to do too much internally, replace the action function with a function that does the same; i.e. replace <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) by <code class="code">function( p, g ) return p^g; end;</code>. The action will be the same, but as the action function is not <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>), the extra processing for special cases is not triggered.</p>
</li>
</ul>
<p><a id="X7FFD731684606BC6" name="X7FFD731684606BC6"></a></p>
<h4>40.5 <span class="Heading">Nice Monomorphisms</span></h4>
<p><strong class="pkg">GAP</strong> contains very efficient algorithms for some special representations of groups (for example pc groups or permutation groups) while for other representations only slow generic methods are available. In this case it can be worthwhile to do all calculations rather in an isomorphic image of the group, which is in a "better" representation. The way to achieve this in <strong class="pkg">GAP</strong> is via <em>nice monomorphisms</em>.</p>
<p>For this mechanism to work, of course there must be effective methods to evaluate the <code class="func">NiceMonomorphism</code> (<a href="chap40.html#X7965086E82ABCF41"><span class="RefLink">40.5-2</span></a>) value on elements and to take preimages under it. As by definition no good algorithms exist for the source group, normally this can only be achieved by using the result of a call to <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>) or <code class="func">GroupHomomorphismByFunction</code> (<a href="chap40.html#X7BC6C20E7CEDBFC5"><span class="RefLink">40.1-4</span></a>) (see also section <a href="chap40.html#X7A121B9E7F78138A"><span class="RefLink">40.3</span></a>).</p>
<p><a id="X78849F81804C44B3" name="X78849F81804C44B3"></a></p>
<h5>40.5-1 IsHandledByNiceMonomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHandledByNiceMonomorphism</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>If this property is <code class="keyw">true</code>, high-valued methods that translate all calculations in <var class="Arg">obj</var> in the image under the <code class="func">NiceMonomorphism</code> (<a href="chap40.html#X7965086E82ABCF41"><span class="RefLink">40.5-2</span></a>) value of <var class="Arg">obj</var> become available for <var class="Arg">obj</var>.</p>
<p><a id="X7965086E82ABCF41" name="X7965086E82ABCF41"></a></p>
<h5>40.5-2 NiceMonomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NiceMonomorphism</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is a homomorphism that is defined (at least) on the whole of <var class="Arg">obj</var> and whose restriction to <var class="Arg">obj</var> is injective. The concrete morphism (and also the image group) will depend on the representation of <var class="Arg">obj</var>.</p>
<p><a id="X7B47BE0983E93A83" name="X7B47BE0983E93A83"></a></p>
<h5>40.5-3 NiceObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NiceObject</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <code class="func">NiceObject</code> value of <var class="Arg">obj</var> is the image of <var class="Arg">obj</var> under the mapping stored as the value of <code class="func">NiceMonomorphism</code> (<a href="chap40.html#X7965086E82ABCF41"><span class="RefLink">40.5-2</span></a>) for <var class="Arg">obj</var>.</p>
<p>A typical example are finite matrix groups, which use a faithful action on vectors to translate all calculations in a permutation group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gl:=GL(3,2);</span>
SL(3,2)
<span class="GAPprompt">gap></span> <span class="GAPinput">IsHandledByNiceMonomorphism(gl);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">NiceObject(gl);</span>
Group([ (5,7)(6,8), (2,3,5)(4,7,6) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Image(NiceMonomorphism(gl),Z(2)*[[1,0,0],[0,1,1],[1,0,1]]);</span>
(2,6)(3,4,7,8)
</pre></div>
<p><a id="X8652149F7F291EE3" name="X8652149F7F291EE3"></a></p>
<h5>40.5-4 IsCanonicalNiceMonomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCanonicalNiceMonomorphism</code>( <var class="Arg">nhom</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A nice monomorphism (see <code class="func">NiceMonomorphism</code> (<a href="chap40.html#X7965086E82ABCF41"><span class="RefLink">40.5-2</span></a>) <var class="Arg">nhom</var> is canonical if the image set will only depend on the set of group elements but not on the generating set and <code class="func">\<</code> (<a href="chap31.html#X7EF67D047F03CA6F"><span class="RefLink">31.11-1</span></a>) comparison of group elements translates through the nice monomorphism. This implies that equal objects will always have equal <code class="func">NiceObject</code> (<a href="chap40.html#X7B47BE0983E93A83"><span class="RefLink">40.5-3</span></a>) values. In some situations however this condition would be expensive to achieve, therefore it is not guaranteed for every nice monomorphism.</p>
<p><a id="X783030917CB43959" name="X783030917CB43959"></a></p>
<h4>40.6 <span class="Heading">Group Automorphisms</span></h4>
<p>Group automorphisms are bijective homomorphism from a group onto itself. An important subclass are automorphisms which are induced by conjugation of the group itself or a supergroup.</p>
<p><a id="X7E52E99487562F3A" name="X7E52E99487562F3A"></a></p>
<h5>40.6-1 ConjugatorIsomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugatorIsomorphism</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a group, and <var class="Arg">g</var> an element in the same family as the elements of <var class="Arg">G</var>. <code class="func">ConjugatorIsomorphism</code> returns the isomorphism from <var class="Arg">G</var> to <code class="code"><var class="Arg">G</var>^<var class="Arg">g</var></code> defined by <span class="SimpleMath">h ↦ h^<var class="Arg">g</var></span> for all <span class="SimpleMath">h ∈ <var class="Arg">G</var></span>.</p>
<p>If <var class="Arg">g</var> normalizes <var class="Arg">G</var> then <code class="func">ConjugatorIsomorphism</code> does the same as <code class="func">ConjugatorAutomorphismNC</code> (<a href="chap40.html#X79ED68CF8139F08A"><span class="RefLink">40.6-2</span></a>).</p>
<p><a id="X79ED68CF8139F08A" name="X79ED68CF8139F08A"></a></p>
<h5>40.6-2 ConjugatorAutomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugatorAutomorphism</code>( <var class="Arg">G</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">‣ ConjugatorAutomorphismNC</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a group, and <var class="Arg">g</var> an element in the same family as the elements of <var class="Arg">G</var> such that <var class="Arg">g</var> normalizes <var class="Arg">G</var>. <code class="func">ConjugatorAutomorphism</code> returns the automorphism of <var class="Arg">G</var> defined by <span class="SimpleMath">h ↦ h^<var class="Arg">g</var></span> for all <span class="SimpleMath">h ∈ <var class="Arg">G</var></span>.</p>
<p>If conjugation by <var class="Arg">g</var> does <em>not</em> leave <var class="Arg">G</var> invariant, <code class="func">ConjugatorAutomorphism</code> returns <code class="keyw">fail</code>; in this case, the isomorphism from <var class="Arg">G</var> to <code class="code"><var class="Arg">G</var>^<var class="Arg">g</var></code> induced by conjugation with <var class="Arg">g</var> can be constructed with <code class="func">ConjugatorIsomorphism</code> (<a href="chap40.html#X7E52E99487562F3A"><span class="RefLink">40.6-1</span></a>).</p>
<p><code class="func">ConjugatorAutomorphismNC</code> does the same as <code class="func">ConjugatorAutomorphism</code>, except that the check is omitted whether <var class="Arg">g</var> normalizes <var class="Arg">G</var> and it is assumed that <var class="Arg">g</var> is chosen to be in <var class="Arg">G</var> if possible.</p>
<p><a id="X7E937A947856D9DA" name="X7E937A947856D9DA"></a></p>
<h5>40.6-3 InnerAutomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InnerAutomorphism</code>( <var class="Arg">G</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">‣ InnerAutomorphismNC</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a group, and <span class="SimpleMath"><var class="Arg">g</var> ∈ <var class="Arg">G</var></span>. <code class="func">InnerAutomorphism</code> returns the automorphism of <var class="Arg">G</var> defined by <span class="SimpleMath">h ↦ h^<var class="Arg">g</var></span> for all <span class="SimpleMath">h ∈ <var class="Arg">G</var></span>.</p>
<p>If <var class="Arg">g</var> is <em>not</em> an element of <var class="Arg">G</var>, <code class="func">InnerAutomorphism</code> returns <code class="keyw">fail</code>; in this case, the isomorphism from <var class="Arg">G</var> to <code class="code"><var class="Arg">G</var>^<var class="Arg">g</var></code> induced by conjugation with <var class="Arg">g</var> can be constructed with <code class="func">ConjugatorIsomorphism</code> (<a href="chap40.html#X7E52E99487562F3A"><span class="RefLink">40.6-1</span></a>) or with <code class="func">ConjugatorAutomorphism</code> (<a href="chap40.html#X79ED68CF8139F08A"><span class="RefLink">40.6-2</span></a>).</p>
<p><code class="func">InnerAutomorphismNC</code> does the same as <code class="func">InnerAutomorphism</code>, except that the check is omitted whether <span class="SimpleMath"><var class="Arg">g</var> ∈ <var class="Arg">G</var></span>.</p>
<p><a id="X7F31FECC7A3D4A8A" name="X7F31FECC7A3D4A8A"></a></p>
<h5>40.6-4 IsConjugatorIsomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsConjugatorIsomorphism</code>( <var class="Arg">hom</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsConjugatorAutomorphism</code>( <var class="Arg">hom</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInnerAutomorphism</code>( <var class="Arg">hom</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Let <var class="Arg">hom</var> be a group general mapping (see <code class="func">IsGroupGeneralMapping</code> (<a href="chap32.html#X819DD174829BF3AE"><span class="RefLink">32.9-4</span></a>)) with source <span class="SimpleMath">G</span>, say. <code class="func">IsConjugatorIsomorphism</code> returns <code class="keyw">true</code> if <var class="Arg">hom</var> is induced by conjugation of <span class="SimpleMath">G</span> by an element <span class="SimpleMath">g</span> that lies in <span class="SimpleMath">G</span> or in a group into which <span class="SimpleMath">G</span> is naturally embedded in the sense described below, and <code class="keyw">false</code> otherwise.</p>
<p>Natural embeddings are dealt with in the case that <span class="SimpleMath">G</span> is a permutation group (see Chapter <a href="chap43.html#X85ED46007CED6191"><span class="RefLink">43</span></a>), a matrix group (see Chapter <a href="chap44.html#X7CF51CB48610A07D"><span class="RefLink">44</span></a>), a finitely presented group (see Chapter <a href="chap47.html#X7AA982637E90B35A"><span class="RefLink">47</span></a>), or a group given w.r.t. a polycyclic presentation (see Chapter <a href="chap46.html#X7EAD57C97EBF7E67"><span class="RefLink">46</span></a>). In all other cases, <code class="func">IsConjugatorIsomorphism</code> may return <code class="keyw">false</code> if <var class="Arg">hom</var> is induced by conjugation but is not an inner automorphism.</p>
<p>If <code class="func">IsConjugatorIsomorphism</code> returns <code class="keyw">true</code> for <var class="Arg">hom</var> then an element <span class="SimpleMath">g</span> that induces <var class="Arg">hom</var> can be accessed as value of the attribute <code class="func">ConjugatorOfConjugatorIsomorphism</code> (<a href="chap40.html#X78FE7E307E86525A"><span class="RefLink">40.6-5</span></a>).</p>
<p><code class="func">IsConjugatorAutomorphism</code> returns <code class="keyw">true</code> if <var class="Arg">hom</var> is an automorphism (see <code class="func">IsEndoGeneralMapping</code> (<a href="chap32.html#X81CFF5F87BBEA8AD"><span class="RefLink">32.13-3</span></a>)) that is regarded as a conjugator isomorphism by <code class="func">IsConjugatorIsomorphism</code>, and <code class="keyw">false</code> otherwise.</p>
<p><code class="func">IsInnerAutomorphism</code> returns <code class="keyw">true</code> if <var class="Arg">hom</var> is a conjugator automorphism such that an element <span class="SimpleMath">g</span> inducing <var class="Arg">hom</var> can be chosen in <span class="SimpleMath">G</span>, and <code class="keyw">false</code> otherwise.</p>
<p><a id="X78FE7E307E86525A" name="X78FE7E307E86525A"></a></p>
<h5>40.6-5 ConjugatorOfConjugatorIsomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugatorOfConjugatorIsomorphism</code>( <var class="Arg">hom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a conjugator isomorphism <var class="Arg">hom</var> (see <code class="func">ConjugatorIsomorphism</code> (<a href="chap40.html#X7E52E99487562F3A"><span class="RefLink">40.6-1</span></a>)), <code class="func">ConjugatorOfConjugatorIsomorphism</code> returns an element <span class="SimpleMath">g</span> such that mapping under <var class="Arg">hom</var> is induced by conjugation with <span class="SimpleMath">g</span>.</p>
<p>To avoid problems with <code class="func">IsInnerAutomorphism</code> (<a href="chap40.html#X7F31FECC7A3D4A8A"><span class="RefLink">40.6-4</span></a>), it is guaranteed that the conjugator is taken from the source of <var class="Arg">hom</var> if possible.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">hgens:=[(1,2,3),(1,2,4)];;h:=Group(hgens);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=GroupHomomorphismByImages(h,h,hgens,[(1,2,3),(2,3,4)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInnerAutomorphism(hom);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ConjugatorOfConjugatorIsomorphism(hom);</span>
(1,2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=GroupHomomorphismByImages(h,h,hgens,[(1,3,2),(1,4,2)]);</span>
[ (1,2,3), (1,2,4) ] -> [ (1,3,2), (1,4,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInnerAutomorphism(hom);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsConjugatorAutomorphism(hom);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ConjugatorOfConjugatorIsomorphism(hom);</span>
(1,2)
</pre></div>
<p><a id="X79640F3682BDBFC1" name="X79640F3682BDBFC1"></a></p>
<h4>40.7 <span class="Heading">Groups of Automorphisms</span></h4>
<p>Group automorphism can be multiplied and inverted and thus it is possible to form groups of automorphisms.</p>
<p><a id="X87677B0787B4461A" name="X87677B0787B4461A"></a></p>
<h5>40.7-1 AutomorphismGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the full automorphism group of the group <var class="Arg">G</var>. The automorphisms act on <var class="Arg">G</var> by the caret operator <code class="code">^</code>. The automorphism group often stores a <code class="func">NiceMonomorphism</code> (<a href="chap40.html#X7965086E82ABCF41"><span class="RefLink">40.5-2</span></a>) value whose image is a permutation group, obtained by the action on a subset of <var class="Arg">G</var>.</p>
<p>Note that current methods for the calculation of the automorphism group of a group <span class="SimpleMath">G</span> require <span class="SimpleMath">G</span> to be a permutation group or a pc group to be efficient. For groups in other representations the calculation is likely very slow.</p>
<p>Also, the <strong class="pkg">AutPGrp</strong> package installs enhanced methods for <code class="func">AutomorphismGroup</code> for finite <span class="SimpleMath">p</span>-groups, and the <strong class="pkg">FGA</strong> package - for finitely generated subgroups of free groups.</p>
<p>Methods may be installed for <code class="func">AutomorphismGroup</code> for other domains, such as e.g. for linear codes in the <strong class="pkg">GUAVA</strong> package, loops in the <strong class="pkg">loops</strong> package and nilpotent Lie algebras in the <strong class="pkg">Sophus</strong> package (see package manuals for their descriptions).</p>
<p><a id="X7FC631B786C1DC8B" name="X7FC631B786C1DC8B"></a></p>
<h5>40.7-2 IsGroupOfAutomorphisms</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGroupOfAutomorphisms</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>indicates whether <var class="Arg">G</var> consists of automorphisms of another group <span class="SimpleMath">H</span>, say. The group <span class="SimpleMath">H</span> can be obtained from <var class="Arg">G</var> via the attribute <code class="func">AutomorphismDomain</code> (<a href="chap40.html#X7B767B9D827EB0FC"><span class="RefLink">40.7-3</span></a>).</p>
<p><a id="X7B767B9D827EB0FC" name="X7B767B9D827EB0FC"></a></p>
<h5>40.7-3 AutomorphismDomain</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismDomain</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">G</var> consists of automorphisms of <span class="SimpleMath">H</span>, this attribute returns <span class="SimpleMath">H</span>.</p>
<p><a id="X7F87D5957D9B991E" name="X7F87D5957D9B991E"></a></p>
<h5>40.7-4 IsAutomorphismGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAutomorphismGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>indicates whether <var class="Arg">G</var> is the full automorphism group of another group <span class="SimpleMath">H</span>, this group is given as <code class="func">AutomorphismDomain</code> (<a href="chap40.html#X7B767B9D827EB0FC"><span class="RefLink">40.7-3</span></a>) value of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,3));</span>
Group([ (1,2,3,4), (1,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">au:=AutomorphismGroup(g);</span>
<group of size 8 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(au);</span>
[ Pcgs([ (2,4), (1,2,3,4), (1,3)(2,4) ]) ->
[ (1,2)(3,4), (1,2,3,4), (1,3)(2,4) ],
Pcgs([ (2,4), (1,2,3,4), (1,3)(2,4) ]) ->
[ (1,3), (1,2,3,4), (1,3)(2,4) ],
Pcgs([ (2,4), (1,2,3,4), (1,3)(2,4) ]) ->
[ (2,4), (1,4,3,2), (1,3)(2,4) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">NiceObject(au);</span>
Group([ (1,2,3,4), (1,3)(2,4), (2,4) ])
</pre></div>
<p><a id="X8476738A7BF9BADA" name="X8476738A7BF9BADA"></a></p>
<h5>40.7-5 InnerAutomorphismsAutomorphismGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InnerAutomorphismsAutomorphismGroup</code>( <var class="Arg">autgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For an automorphism group <var class="Arg">autgroup</var> of a group this attribute stores the subgroup of inner automorphisms (automorphisms induced by conjugation) of the original group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">InnerAutomorphismsAutomorphismGroup(au);</span>
<group with 2 generators>
</pre></div>
<p><a id="X7FC9B6EA7CAADC0A" name="X7FC9B6EA7CAADC0A"></a></p>
<h5>40.7-6 InducedAutomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InducedAutomorphism</code>( <var class="Arg">epi</var>, <var class="Arg">aut</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">aut</var> be an automorphism of a group <span class="SimpleMath">G</span> and <var class="Arg">epi</var> be a homomorphism from <span class="SimpleMath">G</span> to a group <span class="SimpleMath">H</span> such that the kernel of <var class="Arg">epi</var> is fixed under <var class="Arg">aut</var>. Let <span class="SimpleMath">U</span> be the image of <var class="Arg">epi</var>. This command returns the automorphism of <span class="SimpleMath">U</span> induced by <var class="Arg">aut</var> via <var class="Arg">epi</var>, that is, the automorphism of <span class="SimpleMath">U</span> which maps <span class="SimpleMath">g</span><code class="code">^<var class="Arg">epi</var></code> to <code class="code">(</code><span class="SimpleMath">g</span><code class="code">^<var class="Arg">aut</var>)^<var class="Arg">epi</var></code>, for <span class="SimpleMath">g ∈ G</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));</span>
Group([ (1,2,3,4), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=Subgroup(g,[(1,2)(3,4),(1,3)(2,4)]);</span>
Group([ (1,2)(3,4), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">epi:=NaturalHomomorphismByNormalSubgroup(g,n);</span>
[ (1,2,3,4), (1,2) ] -> [ f1*f2, f1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">aut:=InnerAutomorphism(g,(1,2,3));</span>
^(1,2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">InducedAutomorphism(epi,aut);</span>
^f2
</pre></div>
<p><a id="X7A8E961C7F1A57B3" name="X7A8E961C7F1A57B3"></a></p>
<h4>40.8 <span class="Heading">Calculating with Group Automorphisms</span></h4>
<p>Usually the best way to calculate in a group of automorphisms is to translate all calculations to an isomorphic group in a representation, for which better algorithms are available, say a permutation group. This translation can be done automatically using <code class="func">NiceMonomorphism</code> (<a href="chap40.html#X7965086E82ABCF41"><span class="RefLink">40.5-2</span></a>).</p>
<p>Once a group knows to be a group of automorphisms (this can be achieved by testing or setting the property <code class="func">IsGroupOfAutomorphisms</code> (<a href="chap40.html#X7FC631B786C1DC8B"><span class="RefLink">40.7-2</span></a>)), <strong class="pkg">GAP</strong> will try itself to find such a nice monomorphism once calculations in the automorphism group are done.</p>
<p>Note that nice homomorphisms inherit down to subgroups, but cannot necessarily be extended from a subgroup to the whole group. Thus when working with a group of automorphisms, it can be beneficial to enforce calculation of the nice monomorphism for the whole group (for example by explicitly calling <code class="func">Random</code> (<a href="chap30.html#X7FF906E57D6936F8"><span class="RefLink">30.7-1</span></a>) and ignoring the result –it will be stored internally) at the start of the calculation. Otherwise <strong class="pkg">GAP</strong> might first calculate a nice monomorphism for the subgroup, only to be forced to calculate a new nice monomorphism for the whole group later on.</p>
<p>If a good domain for a faithful permutation action is known already, a homomorphism for the action on it can be created using <code class="func">NiceMonomorphismAutomGroup</code> (<a href="chap40.html#X7C9FB0A57BFF6CC0"><span class="RefLink">40.8-2</span></a>). It might be stored by <code class="code">SetNiceMonomorphism</code> (see <code class="func">NiceMonomorphism</code> (<a href="chap40.html#X7965086E82ABCF41"><span class="RefLink">40.5-2</span></a>)).</p>
<p>Another nice way of representing automorphisms as permutations has been described in <a href="chapBib.html#biBSims97">[Sim97]</a>. It is not yet available in <strong class="pkg">GAP</strong>, a description however can be found in section <a href="chap87.html#X7CA84E967B053C2C"><span class="RefLink">87.3</span></a>.</p>
<p><a id="X85691E8386107403" name="X85691E8386107403"></a></p>
<h5>40.8-1 AssignNiceMonomorphismAutomorphismGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssignNiceMonomorphismAutomorphismGroup</code>( <var class="Arg">autgrp</var>, <var class="Arg">group</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes a nice monomorphism for <var class="Arg">autgroup</var> acting on <var class="Arg">group</var> and stores it as <code class="func">NiceMonomorphism</code> (<a href="chap40.html#X7965086E82ABCF41"><span class="RefLink">40.5-2</span></a>) value of <var class="Arg">autgrp</var>.</p>
<p>If the centre of <code class="func">AutomorphismDomain</code> (<a href="chap40.html#X7B767B9D827EB0FC"><span class="RefLink">40.7-3</span></a>) of <var class="Arg">autgrp</var> is trivial, the operation will first try to represent all automorphisms by conjugation (in <var class="Arg">group</var> or in a natural parent of <var class="Arg">group</var>).</p>
<p>If this fails the operation tries to find a small subset of <var class="Arg">group</var> on which the action will be faithful.</p>
<p>The operation sets the attribute <code class="func">NiceMonomorphism</code> (<a href="chap40.html#X7965086E82ABCF41"><span class="RefLink">40.5-2</span></a>) and does not return a value.</p>
<p><a id="X7C9FB0A57BFF6CC0" name="X7C9FB0A57BFF6CC0"></a></p>
<h5>40.8-2 NiceMonomorphismAutomGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NiceMonomorphismAutomGroup</code>( <var class="Arg">autgrp</var>, <var class="Arg">elms</var>, <var class="Arg">elmsgens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function creates a monomorphism for an automorphism group <var class="Arg">autgrp</var> of a group by permuting the group elements in the list <var class="Arg">elms</var>. This list must be chosen to yield a faithful representation. <var class="Arg">elmsgens</var> is a list of generators which are a subset of <var class="Arg">elms</var>. (They can differ from the group's original generators.) It does not yet assign it as <code class="func">NiceMonomorphism</code> (<a href="chap40.html#X7965086E82ABCF41"><span class="RefLink">40.5-2</span></a>) value.</p>
<p><a id="X81B79CC27F47D429" name="X81B79CC27F47D429"></a></p>
<h4>40.9 <span class="Heading">Searching for Homomorphisms</span></h4>
<p><a id="X7B536A32827788C6" name="X7B536A32827788C6"></a></p>
<h5>40.9-1 IsomorphismGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismGroups</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes an isomorphism between the groups <var class="Arg">G</var> and <var class="Arg">H</var> if they are isomorphic and returns <code class="keyw">fail</code> otherwise.</p>
<p>With the existing methods the amount of time needed grows with the size of a generating system of <var class="Arg">G</var>. (Thus in particular for <span class="SimpleMath">p</span>-groups calculations can be slow.) If you do only need to know whether groups are isomorphic, you might want to consider <code class="func">IdGroup</code> (<a href="../../pkg/SmallGrp/doc/chap1.html#X83044B9D7E3BDF35"><span class="RefLink">smallgrp: IdGroup</span></a>) or the random isomorphism test (see <code class="func">RandomIsomorphismTest</code> (<a href="chap46.html#X84F6F9787CB2CF16"><span class="RefLink">46.10-1</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,3));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Group((1,4,6,7)(2,3,5,8), (1,5)(2,6)(3,4)(7,8));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismGroups(g,h);</span>
[ (1,2,3,4), (1,3) ] -> [ (1,4,6,7)(2,3,5,8), (1,2)(3,7)(4,8)(5,6) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismGroups(g,Group((1,2,3,4),(1,2)));</span>
fail
</pre></div>
<p><a id="X7D0C3D5E864CE954" name="X7D0C3D5E864CE954"></a></p>
<h5>40.9-2 AllHomomorphismClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllHomomorphismClasses</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For two groups <var class="Arg">G</var> and <var class="Arg">H</var>, this function returns representatives of all homomorphisms <span class="SimpleMath"><var class="Arg">G</var> to <var class="Arg">H</var></span> up to <var class="Arg">H</var>-conjugacy.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllHomomorphismClasses(SymmetricGroup(4),SymmetricGroup(3)); </span>
[ [ (1,2,3), (2,4) ] -> [ (), () ],
[ (1,2,3), (2,4) ] -> [ (), (1,2) ],
[ (1,2,3), (2,4) ] -> [ (1,2,3), (1,2) ] ]
</pre></div>
<p><a id="X791D12B7845610CE" name="X791D12B7845610CE"></a></p>
<h5>40.9-3 AllHomomorphisms</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllHomomorphisms</code>( <var class="Arg">G</var>, <var class="Arg">H</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">‣ AllEndomorphisms</code>( <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">‣ AllAutomorphisms</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For two groups <var class="Arg">G</var> and <var class="Arg">H</var>, this function returns all homomorphisms <span class="SimpleMath"><var class="Arg">G</var> to <var class="Arg">H</var></span>. Since this number will grow quickly, <code class="func">AllHomomorphismClasses</code> (<a href="chap40.html#X7D0C3D5E864CE954"><span class="RefLink">40.9-2</span></a>) should be used in most cases. <code class="func">AllEndomorphisms</code> returns all homomorphisms from <var class="Arg">G</var> to itself, <code class="func">AllAutomorphisms</code> returns all bijective endomorphisms.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllHomomorphisms(SymmetricGroup(3),SymmetricGroup(3));</span>
[ [ (1,2,3), (1,2) ] -> [ (), () ],
[ (1,2,3), (1,2) ] -> [ (), (1,2) ],
[ (1,2,3), (1,2) ] -> [ (), (2,3) ],
[ (1,2,3), (1,2) ] -> [ (), (1,3) ],
[ (1,2,3), (1,2) ] -> [ (1,2,3), (1,2) ],
[ (1,2,3), (1,2) ] -> [ (1,2,3), (2,3) ],
[ (1,2,3), (1,2) ] -> [ (1,3,2), (1,2) ],
[ (1,2,3), (1,2) ] -> [ (1,2,3), (1,3) ],
[ (1,2,3), (1,2) ] -> [ (1,3,2), (1,3) ],
[ (1,2,3), (1,2) ] -> [ (1,3,2), (2,3) ] ]
</pre></div>
<p><a id="X790C261184EEAB94" name="X790C261184EEAB94"></a></p>
<h5>40.9-4 GQuotients</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GQuotients</code>( <var class="Arg">F</var>, <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes all epimorphisms from <var class="Arg">F</var> onto <var class="Arg">G</var> up to automorphisms of <var class="Arg">G</var>. This classifies all factor groups of <var class="Arg">F</var> which are isomorphic to <var class="Arg">G</var>.</p>
<p>With the existing methods the amount of time needed grows with the size of a generating system of <var class="Arg">G</var>. (Thus in particular for <span class="SimpleMath">p</span>-groups calculations can be slow.)</p>
<p>If the <code class="code">findall</code> option is set to <code class="keyw">false</code>, the algorithm will stop once one homomorphism has been found (this can be faster and might be sufficient if not all homomorphisms are needed).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));</span>
Group([ (1,2,3,4), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Group((1,2,3),(1,2));</span>
Group([ (1,2,3), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">quo:=GQuotients(g,h);</span>
[ [ (1,2,3,4), (1,3,4) ] -> [ (2,3), (1,2,3) ] ]
</pre></div>
<p><a id="X83B417BE7C508DC4" name="X83B417BE7C508DC4"></a></p>
<h5>40.9-5 IsomorphicSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphicSubgroups</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes all monomorphisms from <var class="Arg">H</var> into <var class="Arg">G</var> up to <var class="Arg">G</var>-conjugacy of the image groups. This classifies all <var class="Arg">G</var>-classes of subgroups of <var class="Arg">G</var> which are isomorphic to <var class="Arg">H</var>.</p>
<p>With the existing methods, the amount of time needed grows with the size of a generating system of <var class="Arg">G</var>. (Thus in particular for <span class="SimpleMath">p</span>-groups calculations can be slow.) A main use of <code class="func">IsomorphicSubgroups</code> therefore is to find nonsolvable subgroups (which often can be generated by 2 elements).</p>
<p>(To find <span class="SimpleMath">p</span>-subgroups it is often faster to compute the subgroup lattice of the Sylow subgroup and to use <code class="func">IdGroup</code> (<a href="../../pkg/SmallGrp/doc/chap1.html#X83044B9D7E3BDF35"><span class="RefLink">smallgrp: IdGroup</span></a>) to identify the type of the subgroups.)</p>
<p>If the <code class="code">findall</code> option is set to <code class="keyw">false</code>, the algorithm will stop once one homomorphism has been found (this can be faster and might be sufficient if not all homomorphisms are needed).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));</span>
Group([ (1,2,3,4), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Group((3,4),(1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">emb:=IsomorphicSubgroups(g,h);</span>
[ [ (3,4), (1,2) ] -> [ (1,2), (3,4) ],
[ (3,4), (1,2) ] -> [ (1,3)(2,4), (1,2)(3,4) ] ]
</pre></div>
<p><a id="X7AABA9A27E30BF2B" name="X7AABA9A27E30BF2B"></a></p>
<h5>40.9-6 MorClassLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MorClassLoop</code>( <var class="Arg">range</var>, <var class="Arg">classes</var>, <var class="Arg">params</var>, <var class="Arg">action</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function loops over element tuples taken from <var class="Arg">classes</var> and checks these for properties such as generating a given group, or fulfilling relations. This can be used to find small generating sets or all types of Morphisms. The element tuples are used only up to up to inner automorphisms as all images can be obtained easily from them by conjugation while running through all of them usually would take too long.</p>
<p><var class="Arg">range</var> is a group from which these elements are taken. The classes are given in a list <var class="Arg">classes</var> which is a list of records with the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">classes</code></strong></dt>
<dd><p>list of conjugacy classes</p>
</dd>
<dt><strong class="Mark"><code class="code">representative</code></strong></dt>
<dd><p>One element in the union of these classes</p>
</dd>
<dt><strong class="Mark"><code class="code">size</code></strong></dt>
<dd><p>The sum of the sizes of these classes</p>
</dd>
</dl>
<p><var class="Arg">params</var> is a record containing the following optional components.</p>
<dl>
<dt><strong class="Mark"><code class="code">gens</code></strong></dt>
<dd><p>generators that are to be mapped (for testing morphisms). The length of this list determines the length of element tuples considered.</p>
</dd>
<dt><strong class="Mark"><code class="code">from</code></strong></dt>
<dd><p>a preimage group (that contains <code class="code">gens</code>)</p>
</dd>
<dt><strong class="Mark"><code class="code">to</code></strong></dt>
<dd><p>image group (which might be smaller than <code class="code">range</code>)</p>
</dd>
<dt><strong class="Mark"><code class="code">free</code></strong></dt>
<dd><p>free generators, a list of the same length than the generators <code class="code">gens</code>.</p>
</dd>
<dt><strong class="Mark"><code class="code">rels</code></strong></dt>
<dd><p>some relations that hold among the generators <code class="code">gens</code>. They are given as a list <code class="code">[ word, order ]</code> where <code class="code">word</code> is a word in the free generators <code class="code">free</code>.</p>
</dd>
<dt><strong class="Mark"><code class="code">dom</code></strong></dt>
<dd><p>a set of elements on which automorphisms act faithfully (used to do element tests in partial automorphism groups).</p>
</dd>
<dt><strong class="Mark"><code class="code">aut</code></strong></dt>
<dd><p>Subgroup of already known automorphisms.</p>
</dd>
<dt><strong class="Mark"><code class="code">condition</code></strong></dt>
<dd><p>A function that will be applied to the homomorphism and must return <code class="code">true</code> for the homomorphism to be accepted.</p>
</dd>
</dl>
<p><var class="Arg">action</var> is a number whose bit-representation indicates the requirements which are enforced on the element tuples found, as follows.</p>
<dl>
<dt><strong class="Mark">1</strong></dt>
<dd><p>homomorphism</p>
</dd>
<dt><strong class="Mark">2</strong></dt>
<dd><p>injective</p>
</dd>
<dt><strong class="Mark">4</strong></dt>
<dd><p>surjective</p>
</dd>
<dt><strong class="Mark">8</strong></dt>
<dd><p>find all (otherwise stops after the first find)</p>
</dd>
</dl>
<p>If the search is for homomorphisms, the function returns homomorphisms obtained by mapping the given generators <code class="code">gens</code> instead of element tuples.</p>
<p>The "Morpheus" algorithm used to find homomorphisms is described in <a href="chapBib.html#biBHulpke96">[Hul96, Section V.5]</a>.</p>
<p><a id="X81FC3CEF85CED3DC" name="X81FC3CEF85CED3DC"></a></p>
<h4>40.10 <span class="Heading">Representations for Group Homomorphisms</span></h4>
<p>The different representations of group homomorphisms are used to indicate from what type of group to what type of group they map and thus determine which methods are used to compute images and preimages.</p>
<p>The information in this section is mainly relevant for implementing new methods and not for using homomorphisms.</p>
<p><a id="X82B77A5F7F9EDBC7" name="X82B77A5F7F9EDBC7"></a></p>
<h5>40.10-1 IsGroupGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGroupGeneralMappingByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Representation for mappings from one group to another that are defined by extending a mapping of group generators homomorphically. Instead of record components, the attribute <code class="func">MappingGeneratorsImages</code> (<a href="chap40.html#X863805187A24B5E3"><span class="RefLink">40.10-2</span></a>) is used to store generators and their images.</p>
<p><a id="X863805187A24B5E3" name="X863805187A24B5E3"></a></p>
<h5>40.10-2 MappingGeneratorsImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MappingGeneratorsImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute contains a list of length 2, the first entry being a list of generators of the source of <var class="Arg">map</var> and the second entry a list of their images. This attribute is used, for example, by <code class="func">GroupHomomorphismByImages</code> (<a href="chap40.html#X7F348F497C813BE0"><span class="RefLink">40.1-1</span></a>) to store generators and images.</p>
<p><a id="X7DFBBAB18126B4D9" name="X7DFBBAB18126B4D9"></a></p>
<h5>40.10-3 IsGroupGeneralMappingByAsGroupGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGroupGeneralMappingByAsGroupGeneralMappingByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Representation for mappings that delegate work on a <code class="func">GroupHomomorphismByImages</code> (<a href="chap40.html#X7F348F497C813BE0"><span class="RefLink">40.1-1</span></a>).</p>
<p><a id="X78707DF57C3927EB" name="X78707DF57C3927EB"></a></p>
<h5>40.10-4 IsPreimagesByAsGroupGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPreimagesByAsGroupGeneralMappingByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Representation for mappings that delegate work for preimages to a mapping created with <code class="func">GroupHomomorphismByImages</code> (<a href="chap40.html#X7F348F497C813BE0"><span class="RefLink">40.1-1</span></a>).</p>
<p><a id="X83E10338798F552B" name="X83E10338798F552B"></a></p>
<h5>40.10-5 IsPermGroupGeneralMapping</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPermGroupGeneralMapping</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPermGroupGeneralMappingByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPermGroupHomomorphism</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPermGroupHomomorphismByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>are the representations for mappings that map from a perm group</p>
<p><a id="X83DADD9F7CAD829B" name="X83DADD9F7CAD829B"></a></p>
<h5>40.10-6 IsToPermGroupGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsToPermGroupGeneralMappingByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsToPermGroupHomomorphismByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>is the representation for mappings that map to a perm group</p>
<p><a id="X798E72E77EC85D4A" name="X798E72E77EC85D4A"></a></p>
<h5>40.10-7 IsGroupGeneralMappingByPcgs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGroupGeneralMappingByPcgs</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>is the representations for mappings that map a pcgs to images and thus may use exponents to decompose generators.</p>
<p><a id="X86FF63B784FB8F85" name="X86FF63B784FB8F85"></a></p>
<h5>40.10-8 IsPcGroupGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPcGroupGeneralMappingByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPcGroupHomomorphismByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>is the representation for mappings from a pc group</p>
<p><a id="X79A853B579B250C0" name="X79A853B579B250C0"></a></p>
<h5>40.10-9 IsToPcGroupGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsToPcGroupGeneralMappingByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsToPcGroupHomomorphismByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>is the representation for mappings to a pc group</p>
<p><a id="X7BE2A2EB80DC5CFF" name="X7BE2A2EB80DC5CFF"></a></p>
<h5>40.10-10 IsFromFpGroupGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFromFpGroupGeneralMappingByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFromFpGroupHomomorphismByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>is the representation of mappings from an fp group.</p>
<p><a id="X81090C207F4F6423" name="X81090C207F4F6423"></a></p>
<h5>40.10-11 IsFromFpGroupStdGensGeneralMappingByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFromFpGroupStdGensGeneralMappingByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFromFpGroupStdGensHomomorphismByImages</code>( <var class="Arg">map</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>is the representation of total mappings from an fp group that give images of the standard generators.</p>
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