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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap38.html">[Previous Chapter]</a>    <a href="chap40.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap39_mj.html">[MathJax on]</a></p>
<p><a id="X8716635F7951801B" name="X8716635F7951801B"></a></p>
<div class="ChapSects"><a href="chap39.html#X8716635F7951801B">39 <span class="Heading">Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X822370B47DEA37B1">39.1 <span class="Heading">Group Elements</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X86A022F9800121F8">39.2 <span class="Heading">Creating Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D7B075385435151">39.2-1 <span class="Heading">Group</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F81960287F3E32A">39.2-2 GroupByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8589EF9C7B658B94">39.2-3 GroupWithGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X79C44528864044C5">39.2-4 GeneratorsOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A0747F17B50D967">39.2-5 AsGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E4143A08040BB47">39.2-6 ConjugateGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7939B3177BBD61E4">39.2-7 IsGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X845874BA82E1A11F">39.2-8 InfoGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7BA181CA81D785BB">39.3 <span class="Heading">Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C82AA387A42DCA0">39.3-1 Subgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X842AD37E79CE953E">39.3-2 <span class="Heading">Index (<strong class="pkg">GAP</strong> operation)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8014135884DCC53E">39.3-3 IndexInWholeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7904AC9D7E9A3BB7">39.3-4 AsSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7839D8927E778334">39.3-5 IsSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X838186F9836F678C">39.3-6 IsNormal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8390B5117A10CC52">39.3-7 IsCharacteristicSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X84F5464983655590">39.3-8 ConjugateSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D9990EB837075A4">39.3-9 ConjugateSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82ABF80780CC27AF">39.3-10 IsSubnormal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X829766158665FB54">39.3-11 SubgroupByProperty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E95101F80583E77">39.3-12 SubgroupShell</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7B855B0485C3C6C5">39.4 <span class="Heading">Closures of (Sub)groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D13FC1F8576FFD8">39.4-1 ClosureGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81A20A397C308483">39.4-2 ClosureGroupAddElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82F59F6680D1B0D5">39.4-3 ClosureGroupDefault</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A7AF14A8052F055">39.4-4 ClosureSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7E19F92284F6684E">39.5 <span class="Heading">Expressing Group Elements as Words in Generators</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7FE8A3B08458A1BF">39.5-1 EpimorphismFromFreeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8357294D7B164106">39.5-2 Factorization</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X871508DD808EB487">39.5-3 GrowthFunctionOfGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X87BF1B887C91CA2E">39.6 <span class="Heading">Structure Descriptions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8199B74B84446971">39.6-1 StructureDescription</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X81002AA87DDBC02F">39.7 <span class="Heading">Cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8412ABD57986B9FC">39.7-1 RightCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X835F48248571364F">39.7-2 RightCosets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X85884F177B5D98AE">39.7-3 CanonicalRightCosetElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D7625A1861D9DAB">39.7-4 IsRightCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X78F4F0D8838F5ABF">39.7-5 IsBiCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82F6ABE378B928D1">39.7-6 CosetDecomposition</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X83C723878230D616">39.8 <span class="Heading">Transversals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X85C65D06822E716F">39.8-1 RightTransversal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X78B98B257E981046">39.9 <span class="Heading">Double Cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E51ED757D17254B">39.9-1 DoubleCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F53DABD79BA4F72">39.9-2 RepresentativesContainedRightCosets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A5EFABB86E6D4D5">39.9-3 DoubleCosets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X85ED464F878EF24C">39.9-4 IsDoubleCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A25B1C886CF8C6A">39.9-5 DoubleCosetRepsAndSizes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X84AE7EE77E5FB30E">39.9-6 InfoCoset</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7D474F8F87E4E5D9">39.10 <span class="Heading">Conjugacy Classes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7B2F207F7F85F5B8">39.10-1 ConjugacyClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X871B570284BBA685">39.10-2 ConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D6ED84C86C2979B">39.10-3 ConjugacyClassesByRandomSearch</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X852B3634789D770E">39.10-4 ConjugacyClassesByOrbits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8733F87B7E4C9903">39.10-5 NrConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BD2A4427B7FE248">39.10-6 RationalClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81E9EF0A811072E8">39.10-7 RationalClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X877691247DE23386">39.10-8 GaloisGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83DD148D7DA2ABA9">39.10-9 <span class="Heading">IsConjugate</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81A92F828400FC8A">39.10-10 NthRootsInGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X804F0F037F06E25E">39.11 <span class="Heading">Normal Structure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87B5370C7DFD401D">39.11-1 <span class="Heading">Normalizer</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C4E00297E37AA44">39.11-2 Core</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7CF497C77B1E8938">39.11-3 PCore</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BDEA0A98720D1BB">39.11-4 NormalClosure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D25E7DC7834A703">39.11-5 NormalIntersection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X811B8A4683DDE1F9">39.11-6 ComplementClassesRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8581F4E77B11C610">39.11-7 InfoComplement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7C39EE3E836D6BC6">39.12 <span class="Heading">Specific and Parametrized Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X829759F67D4247CA">39.12-1 TrivialSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A9A3D5578CE33A0">39.12-2 CommutatorSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7CC17CF179ED7EF2">39.12-3 DerivedSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7B10B58F83DDE56E">39.12-4 CommutatorLength</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X780552B57C30DD8F">39.12-5 FittingSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X788C856C82243274">39.12-6 FrattiniSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81D86CCE84193E4F">39.12-7 PrefrattiniSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83D5C8B8865C85F1">39.12-8 PerfectResiduum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8250D99A830DA832">39.12-9 SolvableRadical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81F647FA83D8854F">39.12-10 Socle</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8440C61080CDAA14">39.12-11 SupersolvableResiduum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X796DA805853FAC90">39.12-12 PRump</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7FF0BBDD80E8F6BF">39.13 <span class="Heading">Sylow Subgroups and Hall Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7AA351308787544C">39.13-1 SylowSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8605F3FE7A3B8E12">39.13-2 SylowComplement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7EDBA19E828CD584">39.13-3 HallSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X832E8E6B8347B13F">39.13-4 SylowSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87A245E180D27147">39.13-5 ComplementSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82FE5DFD84F8A3C6">39.13-6 HallSystem</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X87AF37E980382499">39.14 <span class="Heading">Subgroups characterized by prime powers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F069ACC83DB3374">39.14-1 Omega</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83DB33747F069ACC">39.14-2 Agemo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7B75879B8085120A">39.15 <span class="Heading">Group Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7DA27D338374FD28">39.15-1 IsCyclic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X813C952F80E775D4">39.15-2 IsElementaryAbelian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87D062608719F2CD">39.15-3 IsNilpotentGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E3056237C6A5D43">39.15-4 NilpotencyClassOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8755147280C84DBB">39.15-5 IsPerfectGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X809C78D5877D31DF">39.15-6 IsSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D7456077D3D1B86">39.15-7 IsPolycyclicGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7AADF2E88501B9FF">39.15-8 IsSupersolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83977EB97A8E2290">39.15-9 IsMonomialGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A6685D7819AEC32">39.15-10 IsSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X78CC9764803601E7">39.15-11 IsAlmostSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C1709A986B00F97">39.15-12 IsQuasisimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C6AA6897C4409AC">39.15-13 <span class="Heading">IsomorphismTypeInfoFiniteSimpleGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8492B05B822AC58C">39.15-14 SimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X839CDD8C7AE39FD6">39.15-15 SimpleGroupsIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X872E93F586F54FCE">39.15-16 SmallSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7EB47BF27D8CBF72">39.15-17 AllSmallNonabelianSimpleGroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81E22D07871DF37E">39.15-18 IsFinitelyGeneratedGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8648EDA287829755">39.15-19 IsSubsetLocallyFiniteGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8089F18C810B7E3E">39.15-20 IsPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F232B3F8261CE25">39.15-21 IsPowerfulPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7ED4A14F7A235617">39.15-22 IsRegularPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87356BAA7E9E2142">39.15-23 PrimePGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X863434AD7DDE514B">39.15-24 PClassPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X840A4F937ABF15E1">39.15-25 RankPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81130F9A7CFCF6BF">39.15-26 IsPSolvable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87415A8485FCF510">39.15-27 IsPNilpotent</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7F8264FA796B2B7D">39.16 <span class="Heading">Numerical Group Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X812827937F403300">39.16-1 AbelianInvariants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D44470C7DA59C1C">39.16-2 Exponent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X843E0CCA8351FDF4">39.16-3 EulerianFunction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7AEDEDF67CFED672">39.17 <span class="Heading">Subgroup Series</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BDD116F7833800F">39.17-1 ChiefSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7AC93E977AC9ED58">39.17-2 ChiefSeriesThrough</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8724E15F81B51173">39.17-3 ChiefSeriesUnderAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A0E7A8B8495B79D">39.17-4 SubnormalSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81CDCBD67BC98A5A">39.17-5 CompositionSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82C0D0217ACB2042">39.17-6 DisplayCompositionSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A879948834BD889">39.17-7 DerivedSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A9AA1577CEC891F">39.17-8 DerivedLength</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83F057E5791944D6">39.17-9 <span class="Heading">ElementaryAbelianSeries</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X782BD7A47D6B6503">39.17-10 InvariantElementaryAbelianSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X879D55A67DB42676">39.17-11 LowerCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8428592E8773CD7B">39.17-12 UpperCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7809B7ED792669F3">39.17-13 PCentralSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82A34BD681F24A94">39.17-14 JenningsSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C08A8B77EC09CFF">39.17-15 DimensionsLoewyFactors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X84112774812180DD">39.17-16 AscendingChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C5029EE86D7FC96">39.17-17 IntermediateGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X781661FB78DC83B5">39.17-18 IntermediateSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X783CDAA67BDD8195">39.17-19 StructuralSeriesOfGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X84091B0A7E401E2B">39.18 <span class="Heading">Factor Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X80FC390C7F38A13F">39.18-1 NaturalHomomorphismByNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E6EED0185B27C48">39.18-2 FactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7816FA867BF1B8ED">39.18-3 CommutatorFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BB93B9778C5A0B2">39.18-4 MaximalAbelianQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7FC83E4C783572E7">39.18-5 HasAbelianFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7FAC018680B766B7">39.18-6 HasElementaryAbelianFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X822A3AB27919BC1E">39.18-7 CentralizerModulo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7D8EFB2F85AA24EE">39.19 <span class="Heading">Sets of Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7DDE67C67E871336">39.19-1 ConjugacyClassSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C5BBF487977B8CD">39.19-2 IsConjugacyClassSubgroupsRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E986BF48393113A">39.19-3 ConjugacyClassesSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8486C25380853F9B">39.19-4 ConjugacyClassesMaximalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X798BF55C837DB188">39.19-5 MaximalSubgroupClassReps</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X85DAFB7582A88463">39.19-6 LowIndexSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X80399CD4870FFC4B">39.19-7 AllSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X861CD8DA790D81C2">39.19-8 MaximalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X80237A847E24E6CF">39.19-9 NormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82ECAA427C987318">39.19-10 MaximalNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X86FDD9BA819F5644">39.19-11 MinimalNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A823C5A810910C3">39.19-12 CharacteristicSubgroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7FA267497CFC0550">39.20 <span class="Heading">Subgroup Lattice</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7B104E2C86166188">39.20-1 LatticeSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X78928A3582882BFD">39.20-2 ClassElementLattice</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E5DF287825EE7BA">39.20-3 DotFileLatticeSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X815CDA447C5DB285">39.20-4 MaximalSubgroupsLattice</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8138997C871EDF96">39.20-5 MinimalSupergroupsLattice</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87BE970D7B18E2C5">39.20-6 LowLayerSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87FABD5F87AD2568">39.20-7 ContainedConjugates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X79C3619C849F97B8">39.20-8 ContainingConjugates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8111F50C798B0D76">39.20-9 MinimalFaithfulPermutationDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BA3484E7AE0A0E1">39.20-10 RepresentativesPerfectSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7B2233D180DF77A1">39.20-11 ConjugacyClassesPerfectSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BFE573187B4BEF8">39.20-12 Zuppos</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82C12E2C81963B23">39.20-13 InfoLattice</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X85E613D57F28AEFF">39.21 <span class="Heading">Specific Methods for Subgroup Lattice Computations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X86462A567DDBA6BC">39.21-1 LatticeByCyclicExtension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X78918D83835A0EDF">39.21-2 InvariantSubgroupsElementaryAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7AD7804A803910AC">39.21-3 SubgroupsSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F60BBB8874DFE40">39.21-4 SizeConsiderFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X833C51BD7E7812C4">39.21-5 ExactSizeConsiderFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A2C774B7CFF3E07">39.21-6 InfoPcSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X79F894537D526B61">39.22 <span class="Heading">Special Generating Sets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82FD78AF7F80A0E2">39.22-1 GeneratorsSmallest</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A258CCF79552198">39.22-2 LargestElementGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81D15723804771E2">39.22-3 MinimalGeneratingSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X814DBABC878D5232">39.22-4 SmallGeneratingSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D1574457B152333">39.22-5 IndependentGeneratorsOfAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X86F835DA8264A0CE">39.22-6 IndependentGeneratorExponents</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7CA0B6A27E0BE6B8">39.23 <span class="Heading">1-Cohomology</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X847BEC137A49BAF4">39.23-1 <span class="Heading">OneCocycles</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E6438D5834ACCDA">39.23-2 OneCoboundaries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X80400ABD7F40FAA0">39.23-3 OCOneCocycles</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X811E1CF07DABE924">39.23-4 ComplementClassesRepresentativesEA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8199B1D27D487897">39.23-5 InfoCoh</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X80A4B0F282977074">39.24 <span class="Heading">Schur Covers and Multipliers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F619DDA7DD6C43B">39.24-1 EpimorphismSchurCover</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7DD1E37987612042">39.24-2 SchurCover</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X792BC39D7CEB1D27">39.24-3 AbelianInvariantsMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X819E8AEC835F8CD1">39.24-4 Epicentre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8739CD4686301A0E">39.24-5 NonabelianExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E1C8CD77CDB9F71">39.24-6 EpimorphismNonabelianExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BF8DB3D8300BB3F">39.24-7 IsCentralFactor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F4240CD782B6032">39.24-8 <span class="Heading">Covering groups of symmetric groups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7DDA6BC1824F78FD">39.24-9 BasicSpinRepresentationOfSymmetricGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X844CFFDE80F6AD15">39.24-10 SchurCoverOfSymmetricGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E0F4896795E34FC">39.24-11 DoubleCoverOfAlternatingGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7BD95B8D879B73A3">39.25 <span class="Heading">2-Cohomology</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A1EBC3A7AB0D614">39.25-1 TwoCohomologyGeneric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A65366879BB3977">39.25-2 FpGroupCocycle</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X865722987E0E19B6">39.26 <span class="Heading">Tests for the Availability of Methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X798F13EA810FB215">39.26-1 CanEasilyTestMembership</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C2A89607BDFD920">39.26-2 CanEasilyComputeWithIndependentGensAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83245C82835D496C">39.26-3 CanComputeSize</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8268965487364912">39.26-4 CanComputeSizeAnySubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82DDE00D82A32083">39.26-5 CanComputeIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BE7C36B84C23511">39.26-6 CanComputeIsSubset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87D62C2C7C375E2D">39.26-7 KnowsHowToDecompose</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X83A9997586694DC0">39.27 <span class="Heading">Specific functions for Normalizer calculation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X84ABCA997D294B36">39.27-1 NormalizerViaRadical</a></span>
</div></div>
</div>

<h3>39 <span class="Heading">Groups</span></h3>

<p>This chapter explains how to create groups and defines operations for groups, that is operations whose definition does not depend on the representation used. However methods for these operations in most cases will make use of the representation.</p>

<p>If not otherwise specified, in all examples in this chapter the group <code class="code">g</code> will be the symmetric group <span class="SimpleMath">S_4</span> acting on the letters <span class="SimpleMath">{ 1, ..., 4 }</span>.</p>

<p><a id="X822370B47DEA37B1" name="X822370B47DEA37B1"></a></p>

<h4>39.1 <span class="Heading">Group Elements</span></h4>

<p>Groups in <strong class="pkg">GAP</strong> are written multiplicatively. The elements from which a group can be generated must permit multiplication and multiplicative inversion (see <a href="chap31.html#X7B97A0307EA161E5"><span class="RefLink">31.14</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=(1,2,3);;b:=(2,3,4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">One(a);</span>
()
<span class="GAPprompt">gap></span> <span class="GAPinput">Inverse(b);</span>
(2,4,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">a*b;</span>
(1,3)(2,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(a*b);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Order( [ [ 1, 1 ], [ 0, 1 ] ] );</span>
infinity
</pre></div>

<p>The next example may run into an infinite loop because the given matrix in fact has infinite order.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order( [ [ 1, 1 ], [ 0, 1 ] ] * Indeterminate( Rationals ) );</span>
#I  Order: warning, order of <mat> might be infinite
</pre></div>

<p>Since groups are domains, the recommended command to compute the order of a group is <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>). For convenience, group orders can also be computed with <code class="func">Order</code> (<a href="chap31.html#X84F59A2687C62763"><span class="RefLink">31.10-10</span></a>).</p>

<p>The operation <code class="func">Comm</code> (<a href="chap31.html#X80761843831B468E"><span class="RefLink">31.12-3</span></a>) can be used to compute the commutator of two elements, the operation <code class="func">LeftQuotient</code> (<a href="chap31.html#X7A37082878DB3930"><span class="RefLink">31.12-2</span></a>) computes the product <span class="SimpleMath">x^{-1} y</span>.</p>

<p><a id="X86A022F9800121F8" name="X86A022F9800121F8"></a></p>

<h4>39.2 <span class="Heading">Creating Groups</span></h4>

<p>When groups are created from generators, this means that the generators must be elements that can be multiplied and inverted (see also <a href="chap31.html#X82039A218274826F"><span class="RefLink">31.3</span></a>). For creating a free group on a set of symbols, see <code class="func">FreeGroup</code> (<a href="chap37.html#X8215999E835290F0"><span class="RefLink">37.2-1</span></a>).</p>

<p><a id="X7D7B075385435151" name="X7D7B075385435151"></a></p>

<h5>39.2-1 <span class="Heading">Group</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Group</code>( <var class="Arg">gen</var>, <var class="Arg">...</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">‣ Group</code>( <var class="Arg">gens</var>[, <var class="Arg">id</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">Group( <var class="Arg">gen</var>, ... )</code> is the group generated by the arguments <var class="Arg">gen</var>, ...</p>

<p>If the only argument <var class="Arg">gens</var> is a list that is not a matrix then <code class="code">Group( <var class="Arg">gens</var> )</code> is the group generated by the elements of that list.</p>

<p>If there are two arguments, a list <var class="Arg">gens</var> and an element <var class="Arg">id</var>, then <code class="code">Group( <var class="Arg">gens</var>, <var class="Arg">id</var> )</code> is the group generated by the elements of <var class="Arg">gens</var>, with identity <var class="Arg">id</var>.</p>

<p>Note that the value of the attribute <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>) need not be equal to the list <var class="Arg">gens</var> of generators entered as argument. Use <code class="func">GroupWithGenerators</code> (<a href="chap39.html#X8589EF9C7B658B94"><span class="RefLink">39.2-3</span></a>) if you want to be sure that the argument <var class="Arg">gens</var> is stored as value of <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>).</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) ])
</pre></div>

<p><a id="X7F81960287F3E32A" name="X7F81960287F3E32A"></a></p>

<h5>39.2-2 GroupByGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupByGenerators</code>( <var class="Arg">gens</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">‣ GroupByGenerators</code>( <var class="Arg">gens</var>, <var class="Arg">id</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">GroupByGenerators</code> returns the group <span class="SimpleMath">G</span> generated by the list <var class="Arg">gens</var>. If a second argument <var class="Arg">id</var> is present then this is stored as the identity element of the group.</p>

<p>The value of the attribute <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>) of <span class="SimpleMath">G</span> need not be equal to <var class="Arg">gens</var>. <code class="func">GroupByGenerators</code> is the underlying operation called by <code class="func">Group</code> (<a href="chap39.html#X7D7B075385435151"><span class="RefLink">39.2-1</span></a>).</p>

<p><a id="X8589EF9C7B658B94" name="X8589EF9C7B658B94"></a></p>

<h5>39.2-3 GroupWithGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupWithGenerators</code>( <var class="Arg">gens</var>[, <var class="Arg">id</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">GroupWithGenerators</code> returns the group <span class="SimpleMath">G</span> generated by the list <var class="Arg">gens</var>. If a second argument <var class="Arg">id</var> is present then this is stored as the identity element of the group. The value of the attribute <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>) of <span class="SimpleMath">G</span> is equal to <var class="Arg">gens</var>.</p>

<p><a id="X79C44528864044C5" name="X79C44528864044C5"></a></p>

<h5>39.2-4 GeneratorsOfGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of generators of the group <var class="Arg">G</var>. If <var class="Arg">G</var> has been created by the command <code class="func">GroupWithGenerators</code> (<a href="chap39.html#X8589EF9C7B658B94"><span class="RefLink">39.2-3</span></a>) with argument <var class="Arg">gens</var>, then the list returned by <code class="func">GeneratorsOfGroup</code> will be equal to <var class="Arg">gens</var>. For such a group, each generator can also be accessed using the <code class="code">.</code> operator (see <code class="func">GeneratorsOfDomain</code> (<a href="chap31.html#X7E353DD1838AB223"><span class="RefLink">31.9-2</span></a>)): for a positive integer <span class="SimpleMath">i</span>, <code class="code"><var class="Arg">G</var>.i</code> returns the <span class="SimpleMath">i</span>-th element of the list returned by <code class="func">GeneratorsOfGroup</code>. Moreover, if <var class="Arg">G</var> is a free group, and <code class="code">name</code> is the name of a generator of <var class="Arg">G</var> then <code class="code"><var class="Arg">G</var>.name</code> also returns this generator.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=GroupWithGenerators([(1,2,3,4),(1,2)]);</span>
Group([ (1,2,3,4), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(g);</span>
[ (1,2,3,4), (1,2) ]
</pre></div>

<p>While in this example <strong class="pkg">GAP</strong> displays the group via the generating set stored in the attribute <code class="func">GeneratorsOfGroup</code>, the methods installed for <code class="func">View</code> (<a href="chap6.html#X851902C583B84CDC"><span class="RefLink">6.3-3</span></a>) will in general display only some information about the group which may even be just the fact that it is a group.</p>

<p><a id="X7A0747F17B50D967" name="X7A0747F17B50D967"></a></p>

<h5>39.2-5 AsGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsGroup</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>if the elements of the collection <var class="Arg">D</var> form a group the command returns this group, otherwise it returns <code class="keyw">fail</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsGroup([(1,2)]);</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">AsGroup([(),(1,2)]);</span>
Group([ (1,2) ])
</pre></div>

<p><a id="X7E4143A08040BB47" name="X7E4143A08040BB47"></a></p>

<h5>39.2-6 ConjugateGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugateGroup</code>( <var class="Arg">G</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the conjugate group of <var class="Arg">G</var>, obtained by applying the conjugating element <var class="Arg">obj</var>.</p>

<p>To form a conjugate (group) by any object acting via <code class="code">^</code>, one can also use the infix operator <code class="code">^</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ConjugateGroup(g,(1,5));</span>
Group([ (2,3,4,5), (2,5) ])
</pre></div>

<p><a id="X7939B3177BBD61E4" name="X7939B3177BBD61E4"></a></p>

<h5>39.2-7 IsGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGroup</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A group is a magma-with-inverses (see <code class="func">IsMagmaWithInverses</code> (<a href="chap35.html#X82CBFF648574B830"><span class="RefLink">35.1-4</span></a>)) and associative (see <code class="func">IsAssociative</code> (<a href="chap35.html#X7C83B5A47FD18FB7"><span class="RefLink">35.4-7</span></a>)) multiplication.</p>

<p><code class="code">IsGroup</code> tests whether the object <var class="Arg">obj</var> fulfills these conditions, it does <em>not</em> test whether <var class="Arg">obj</var> is a set of elements that forms a group under multiplication; use <code class="func">AsGroup</code> (<a href="chap39.html#X7A0747F17B50D967"><span class="RefLink">39.2-5</span></a>) if you want to perform such a test. (See <a href="chap13.html#X7CC6903E78F24167"><span class="RefLink">13.3</span></a> for details about categories.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGroup(g);</span>
true
</pre></div>

<p><a id="X845874BA82E1A11F" name="X845874BA82E1A11F"></a></p>

<h5>39.2-8 InfoGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoGroup</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>is the info class for the generic group theoretic functions (see <a href="chap7.html#X7A9C902479CB6F7C"><span class="RefLink">7.4</span></a>).</p>

<p><a id="X7BA181CA81D785BB" name="X7BA181CA81D785BB"></a></p>

<h4>39.3 <span class="Heading">Subgroups</span></h4>

<p>For the general concept of parents and subdomains, see <a href="chap31.html#X7CBDD36E7B7BE286"><span class="RefLink">31.7</span></a> and <a href="chap31.html#X7B58FDEF80338DD6"><span class="RefLink">31.8</span></a>. More functions that construct certain subgroups can be found in the sections <a href="chap39.html#X804F0F037F06E25E"><span class="RefLink">39.11</span></a>, <a href="chap39.html#X7C39EE3E836D6BC6"><span class="RefLink">39.12</span></a>, <a href="chap39.html#X7FF0BBDD80E8F6BF"><span class="RefLink">39.13</span></a>, and <a href="chap39.html#X87AF37E980382499"><span class="RefLink">39.14</span></a>.</p>

<p>If a group <span class="SimpleMath">U</span> is created as a subgroup of another group <span class="SimpleMath">G</span>, <span class="SimpleMath">G</span> becomes the parent of <span class="SimpleMath">U</span>. There is no <q>universal</q> parent group, parent-child chains can be arbitrary long. <strong class="pkg">GAP</strong> stores the result of some operations (such as <code class="func">Normalizer</code> (<a href="chap39.html#X87B5370C7DFD401D"><span class="RefLink">39.11-1</span></a>)) with the parent as an attribute.</p>

<p><a id="X7C82AA387A42DCA0" name="X7C82AA387A42DCA0"></a></p>

<h5>39.3-1 Subgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Subgroup</code>( <var class="Arg">G</var>, <var class="Arg">gens</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">‣ SubgroupNC</code>( <var class="Arg">G</var>, <var class="Arg">gens</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">‣ Subgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates the subgroup <var class="Arg">U</var> of <var class="Arg">G</var> generated by <var class="Arg">gens</var>. The <code class="func">Parent</code> (<a href="chap31.html#X7BC856CC7F116BB0"><span class="RefLink">31.7-1</span></a>) value of <var class="Arg">U</var> will be <var class="Arg">G</var>. The <code class="code">NC</code> version does not check, whether the elements in <var class="Arg">gens</var> actually lie in <var class="Arg">G</var>.</p>

<p>The unary version of <code class="func">Subgroup</code> creates a (shell) subgroup that does not even know generators but can be used to collect information about a particular subgroup over time.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:=Subgroup(g,[(1,2,3),(1,2)]);</span>
Group([ (1,2,3), (1,2) ])
</pre></div>

<p><a id="X842AD37E79CE953E" name="X842AD37E79CE953E"></a></p>

<h5>39.3-2 <span class="Heading">Index (<strong class="pkg">GAP</strong> operation)</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Index</code>( <var class="Arg">G</var>, <var class="Arg">U</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">‣ IndexNC</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a subgroup <var class="Arg">U</var> of the group <var class="Arg">G</var>, <code class="func">Index</code> returns the index <span class="SimpleMath">[<var class="Arg">G</var>:<var class="Arg">U</var>] = |<var class="Arg">G</var>| / |<var class="Arg">U</var>|</span> of <var class="Arg">U</var> in <var class="Arg">G</var>. The <code class="code">NC</code> version does not test whether <var class="Arg">U</var> is contained in <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Index(g,u);</span>
4
</pre></div>

<p><a id="X8014135884DCC53E" name="X8014135884DCC53E"></a></p>

<h5>39.3-3 IndexInWholeGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IndexInWholeGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If the family of elements of <var class="Arg">G</var> itself forms a group <var class="Arg">P</var>, this attribute returns the index of <var class="Arg">G</var> in <var class="Arg">P</var>. It is used primarily for free groups or finitely presented groups.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">freegp:=FreeGroup(1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">freesub:=Subgroup(freegp,[freegp.1^5]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IndexInWholeGroup(freesub);</span>
5
</pre></div>

<p><a id="X7904AC9D7E9A3BB7" name="X7904AC9D7E9A3BB7"></a></p>

<h5>39.3-4 AsSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>creates a subgroup of <var class="Arg">G</var> which contains the same elements as <var class="Arg">U</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">v:=AsSubgroup(g,Group((1,2,3),(1,4)));</span>
Group([ (1,2,3), (1,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Parent(v);</span>
Group([ (1,2,3,4), (1,2) ])
</pre></div>

<p><a id="X7839D8927E778334" name="X7839D8927E778334"></a></p>

<h5>39.3-5 IsSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">IsSubgroup</code> returns <code class="keyw">true</code> if <var class="Arg">U</var> is a group that is a subset of the domain <var class="Arg">G</var>. This is actually checked by calling <code class="code">IsGroup( <var class="Arg">U</var> )</code> and <code class="code">IsSubset( <var class="Arg">G</var>, <var class="Arg">U</var> )</code>; note that special methods for <code class="func">IsSubset</code> (<a href="chap30.html#X79CA175481F8105F"><span class="RefLink">30.5-1</span></a>) are available that test only generators of <var class="Arg">U</var> if <var class="Arg">G</var> is closed under the group operations. So in most cases, for example whenever one knows already that <var class="Arg">U</var> is a group, it is better to call only <code class="func">IsSubset</code> (<a href="chap30.html#X79CA175481F8105F"><span class="RefLink">30.5-1</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(g,u);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">v:=Group((1,2,3),(1,2));</span>
Group([ (1,2,3), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">u=v;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(g,v);</span>
true
</pre></div>

<p><a id="X838186F9836F678C" name="X838186F9836F678C"></a></p>

<h5>39.3-6 IsNormal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNormal</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the group <var class="Arg">G</var> normalizes the group <var class="Arg">U</var> and <code class="keyw">false</code> otherwise.</p>

<p>A group <var class="Arg">G</var> <em>normalizes</em> a group <var class="Arg">U</var> if and only if for every <span class="SimpleMath">g ∈ <var class="Arg">G</var></span> and <span class="SimpleMath">u ∈ <var class="Arg">U</var></span> the element <span class="SimpleMath">u^g</span> is a member of <var class="Arg">U</var>. Note that <var class="Arg">U</var> need not be a subgroup of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormal(g,u);</span>
false
</pre></div>

<p><a id="X8390B5117A10CC52" name="X8390B5117A10CC52"></a></p>

<h5>39.3-7 IsCharacteristicSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCharacteristicSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>tests whether <var class="Arg">N</var> is invariant under all automorphisms of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCharacteristicSubgroup(g,u);</span>
false
</pre></div>

<p><a id="X84F5464983655590" name="X84F5464983655590"></a></p>

<h5>39.3-8 ConjugateSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugateSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a group <var class="Arg">G</var> which has a parent group <code class="code">P</code> (see <code class="func">Parent</code> (<a href="chap31.html#X7BC856CC7F116BB0"><span class="RefLink">31.7-1</span></a>)), returns the subgroup of <code class="code">P</code>, obtained by conjugating <var class="Arg">G</var> using the conjugating element <var class="Arg">g</var>.</p>

<p>If <var class="Arg">G</var> has no parent group, it just delegates to the call to <code class="func">ConjugateGroup</code> (<a href="chap39.html#X7E4143A08040BB47"><span class="RefLink">39.2-6</span></a>) with the same arguments.</p>

<p>To form a conjugate (subgroup) by any object acting via <code class="code">^</code>, one can also use the infix operator <code class="code">^</code>.</p>

<p><a id="X7D9990EB837075A4" name="X7D9990EB837075A4"></a></p>

<h5>39.3-9 ConjugateSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugateSubgroups</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a list of all images of the group <var class="Arg">U</var> under conjugation action by <var class="Arg">G</var>.</p>

<p><a id="X82ABF80780CC27AF" name="X82ABF80780CC27AF"></a></p>

<h5>39.3-10 IsSubnormal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubnormal</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A subgroup <var class="Arg">U</var> of the group <var class="Arg">G</var> is subnormal if it is contained in a subnormal series of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubnormal(g,Group((1,2,3)));</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubnormal(g,Group((1,2)(3,4)));</span>
true
</pre></div>

<p><a id="X829766158665FB54" name="X829766158665FB54"></a></p>

<h5>39.3-11 SubgroupByProperty</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubgroupByProperty</code>( <var class="Arg">G</var>, <var class="Arg">prop</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates a subgroup of <var class="Arg">G</var> consisting of those elements fulfilling <var class="Arg">prop</var> (which is a tester function). No test is done whether the property actually defines a subgroup.</p>

<p>Note that currently very little functionality beyond an element test exists for groups created this way.</p>

<p><a id="X7E95101F80583E77" name="X7E95101F80583E77"></a></p>

<h5>39.3-12 SubgroupShell</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubgroupShell</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates a subgroup of <var class="Arg">G</var> which at this point is not yet specified further (but will be later, for example by assigning a generating set).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:=SubgroupByProperty(g,i->3^i=3);</span>
<subgrp of Group([ (1,2,3,4), (1,2) ]) by property>
<span class="GAPprompt">gap></span> <span class="GAPinput">(1,3) in u; (1,4) in u; (1,5) in u;</span>
false
true
false
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(u);</span>
[ (1,2), (1,4,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">u:=SubgroupShell(g);</span>
<group>
</pre></div>

<p><a id="X7B855B0485C3C6C5" name="X7B855B0485C3C6C5"></a></p>

<h4>39.4 <span class="Heading">Closures of (Sub)groups</span></h4>

<p><a id="X7D13FC1F8576FFD8" name="X7D13FC1F8576FFD8"></a></p>

<h5>39.4-1 ClosureGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClosureGroup</code>( <var class="Arg">G</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>creates the group generated by the elements of <var class="Arg">G</var> and <var class="Arg">obj</var>. <var class="Arg">obj</var> can be either an element or a collection of elements, in particular another group.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(24,12);;u:=Subgroup(g,[g.3,g.4]);</span>
Group([ f3, f4 ])
--> --------------------

--> maximum size reached

--> --------------------

100%


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