Quelle chap40.html
Sprache: HTML
|
|
| products/sources/formale Sprachen/GAP/doc/ref/chap40.html |
 |
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ref) - Chapter 40: Group Homomorphisms</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap40" onload="jscontent()">
<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chap41.html">41</a> <a href="chap42.html">42</a> <a href="chap43.html">43</a> <a href="chap44.html">44</a> <a href="chap45.html">45</a> <a href="chap46.html">46</a> <a href="chap47.html">47</a> <a href="chap48.html">48</a> <a href="chap49.html">49</a> <a href="chap50.html">50</a> <a href="chap51.html">51</a> <a href="chap52.html">52</a> <a href="chap53.html">53</a> <a href="chap54.html">54</a> <a href="chap55.html">55</a> <a href="chap56.html">56</a> <a href="chap57.html">57</a> <a href="chap58.html">58</a> <a href="chap59.html">59</a> <a href="chap60.html">60</a> <a href="chap61.html">61</a> <a href="chap62.html">62</a> <a href="chap63.html">63</a> <a href="chap64.html">64</a> <a href="chap65.html">65</a> <a href="chap66.html">66</a> <a href="chap67.html">67</a> <a href="chap68.html">68</a> <a href="chap69.html">69</a> <a href="chap70.html">70</a> <a href="chap71.html">71</a> <a href="chap72.html">72</a> <a href="chap73.html">73</a> <a href="chap74.html">74</a> <a href="chap75.html">75</a> <a href="chap76.html">76</a> <a href="chap77.html">77</a> <a href="chap78.html">78</a> <a href="chap79.html">79</a> <a href="chap80.html">80</a> <a href="chap81.html">81</a> <a href="chap82.html">82</a> <a href="chap83.html">83</a> <a href="chap84.html">84</a> <a href="chap85.html">85</a> <a href="chap86.html">86</a> <a href="chap87.html">87</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<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>
<p id="mathjaxlink" class="pcenter"><a href="chap40_mj.html">[MathJax on]</a></p>
<p><a id="X83702FC27B3C3098" name="X83702FC27B3C3098"></a></p>
<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#X7957AC21782B6C8C">40.7-6 InnerAutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap40.html#X7FC9B6EA7CAADC0A">40.7-7 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 <strong class="pkg">GAP</strong>'s facilities to work with automorphism groups.
<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">39.26-7</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 <q>surjective</q> (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 <q>better</q> 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>It may happen that one knows a monomorphism <code class="code">map</code> that is suitable as a nice monomorphism of a given group <code class="code">G</code>, for example if <code class="code">G</code> is a matrix group and <code class="code">map</code> describes the faithful action of <code class="code">G</code> on a small set of vectors. In such a case, one can prescribe this monomorphism via <code class="code">SetNiceMonomorphism( G, map )</code>, provided that <code class="code">map</code> stores that it is injective. The latter can be achieved by calling <code class="code">IsInjective( map )</code> or, if one is sure that <code class="code">map</code> is injective and one wants to avoid the overhead of the test, by calling <code class="code">SetIsInjective( map, true )</code>.</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>WARNING: The domain of the homomorphism may be larger than <var class="Arg">obj</var>. To obtain the image of <var class="Arg">obj</var> under the homomorphism, use <code class="func">NiceObject</code> (<a href="chap40.html#X7B47BE0983E93A83"><span class="RefLink">40.5-3</span></a>); see there for an example where it matters.</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</ | |
