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<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap38_mj.html">[Previous Chapter]</a> <a href="chap40_mj.html">[Next Chapter]</a> </div>
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<div class="ChapSects"><a href="chap39_mj.html#X8716635F7951801B">39 <span class="Heading">Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.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_mj.html#X7D8E473384DE9CD4">39.2-1 Group</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7F81960287F3E32A">39.2-2 GroupByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8589EF9C7B658B94">39.2-3 GroupWithGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X79C44528864044C5">39.2-4 GeneratorsOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A0747F17B50D967">39.2-5 AsGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7E4143A08040BB47">39.2-6 ConjugateGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7939B3177BBD61E4">39.2-7 IsGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X845874BA82E1A11F">39.2-8 InfoGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7C82AA387A42DCA0">39.3-1 Subgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X8014135884DCC53E">39.3-3 IndexInWholeGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7904AC9D7E9A3BB7">39.3-4 AsSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7839D8927E778334">39.3-5 IsSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X838186F9836F678C">39.3-6 IsNormal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8390B5117A10CC52">39.3-7 IsCharacteristicSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X84F5464983655590">39.3-8 ConjugateSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7D9990EB837075A4">39.3-9 ConjugateSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X82ABF80780CC27AF">39.3-10 IsSubnormal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X829766158665FB54">39.3-11 SubgroupByProperty</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7E95101F80583E77">39.3-12 SubgroupShell</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7D13FC1F8576FFD8">39.4-1 ClosureGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X81A20A397C308483">39.4-2 ClosureGroupAddElm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X82F59F6680D1B0D5">39.4-3 ClosureGroupDefault</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A7AF14A8052F055">39.4-4 ClosureSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7FE8A3B08458A1BF">39.5-1 EpimorphismFromFreeGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8357294D7B164106">39.5-2 Factorization</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X871508DD808EB487">39.5-3 GrowthFunctionOfGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X8199B74B84446971">39.6-1 StructureDescription</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X8412ABD57986B9FC">39.7-1 RightCoset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X835F48248571364F">39.7-2 RightCosets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X85884F177B5D98AE">39.7-3 CanonicalRightCosetElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7D7625A1861D9DAB">39.7-4 IsRightCoset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X78F4F0D8838F5ABF">39.7-5 IsBiCoset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X82F6ABE378B928D1">39.7-6 CosetDecomposition</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X85C65D06822E716F">39.8-1 RightTransversal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7E51ED757D17254B">39.9-1 DoubleCoset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7F53DABD79BA4F72">39.9-2 RepresentativesContainedRightCosets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A5EFABB86E6D4D5">39.9-3 DoubleCosets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X85ED464F878EF24C">39.9-4 IsDoubleCoset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A25B1C886CF8C6A">39.9-5 DoubleCosetRepsAndSizes</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X84AE7EE77E5FB30E">39.9-6 InfoCoset</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7B2F207F7F85F5B8">39.10-1 ConjugacyClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X871B570284BBA685">39.10-2 ConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7D6ED84C86C2979B">39.10-3 ConjugacyClassesByRandomSearch</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X852B3634789D770E">39.10-4 ConjugacyClassesByOrbits</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8733F87B7E4C9903">39.10-5 NrConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7BD2A4427B7FE248">39.10-6 RationalClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X81E9EF0A811072E8">39.10-7 RationalClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X877691247DE23386">39.10-8 GaloisGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X83DD148D7DA2ABA9">39.10-9 <span class="Heading">IsConjugate</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X81A92F828400FC8A">39.10-10 NthRootsInGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X87B5370C7DFD401D">39.11-1 <span class="Heading">Normalizer</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7C4E00297E37AA44">39.11-2 Core</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7CF497C77B1E8938">39.11-3 PCore</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7BDEA0A98720D1BB">39.11-4 NormalClosure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7D25E7DC7834A703">39.11-5 NormalIntersection</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X811B8A4683DDE1F9">39.11-6 ComplementClassesRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8581F4E77B11C610">39.11-7 InfoComplement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X829759F67D4247CA">39.12-1 TrivialSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A9A3D5578CE33A0">39.12-2 CommutatorSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7CC17CF179ED7EF2">39.12-3 DerivedSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7B10B58F83DDE56E">39.12-4 CommutatorLength</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X780552B57C30DD8F">39.12-5 FittingSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X788C856C82243274">39.12-6 FrattiniSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X81D86CCE84193E4F">39.12-7 PrefrattiniSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X83D5C8B8865C85F1">39.12-8 PerfectResiduum</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X787F5F14844FAACE">39.12-9 RadicalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X81F647FA83D8854F">39.12-10 Socle</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8440C61080CDAA14">39.12-11 SupersolvableResiduum</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X796DA805853FAC90">39.12-12 PRump</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7AA351308787544C">39.13-1 SylowSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8605F3FE7A3B8E12">39.13-2 SylowComplement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7EDBA19E828CD584">39.13-3 HallSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X832E8E6B8347B13F">39.13-4 SylowSystem</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X87A245E180D27147">39.13-5 ComplementSystem</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X82FE5DFD84F8A3C6">39.13-6 HallSystem</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7F069ACC83DB3374">39.14-1 Omega</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X83DB33747F069ACC">39.14-2 Agemo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7DA27D338374FD28">39.15-1 IsCyclic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X813C952F80E775D4">39.15-2 IsElementaryAbelian</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X87D062608719F2CD">39.15-3 IsNilpotentGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7E3056237C6A5D43">39.15-4 NilpotencyClassOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8755147280C84DBB">39.15-5 IsPerfectGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X809C78D5877D31DF">39.15-6 IsSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7D7456077D3D1B86">39.15-7 IsPolycyclicGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7AADF2E88501B9FF">39.15-8 IsSupersolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X83977EB97A8E2290">39.15-9 IsMonomialGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A6685D7819AEC32">39.15-10 IsSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X78CC9764803601E7">39.15-11 IsAlmostSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7C6AA6897C4409AC">39.15-12 <span class="Heading">IsomorphismTypeInfoFiniteSimpleGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8492B05B822AC58C">39.15-13 SimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X839CDD8C7AE39FD6">39.15-14 SimpleGroupsIterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X872E93F586F54FCE">39.15-15 SmallSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7EB47BF27D8CBF72">39.15-16 AllSmallNonabelianSimpleGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X81E22D07871DF37E">39.15-17 IsFinitelyGeneratedGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8648EDA287829755">39.15-18 IsSubsetLocallyFiniteGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8089F18C810B7E3E">39.15-19 IsPGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7F232B3F8261CE25">39.15-20 IsPowerfulPGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X87356BAA7E9E2142">39.15-21 PrimePGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X863434AD7DDE514B">39.15-22 PClassPGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X840A4F937ABF15E1">39.15-23 RankPGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X81130F9A7CFCF6BF">39.15-24 IsPSolvable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X87415A8485FCF510">39.15-25 IsPNilpotent</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X812827937F403300">39.16-1 AbelianInvariants</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7D44470C7DA59C1C">39.16-2 Exponent</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X843E0CCA8351FDF4">39.16-3 EulerianFunction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7BDD116F7833800F">39.17-1 ChiefSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7AC93E977AC9ED58">39.17-2 ChiefSeriesThrough</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8724E15F81B51173">39.17-3 ChiefSeriesUnderAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A0E7A8B8495B79D">39.17-4 SubnormalSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X81CDCBD67BC98A5A">39.17-5 CompositionSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X82C0D0217ACB2042">39.17-6 DisplayCompositionSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A879948834BD889">39.17-7 DerivedSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A9AA1577CEC891F">39.17-8 DerivedLength</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X83F057E5791944D6">39.17-9 <span class="Heading">ElementaryAbelianSeries</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X782BD7A47D6B6503">39.17-10 InvariantElementaryAbelianSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X879D55A67DB42676">39.17-11 LowerCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8428592E8773CD7B">39.17-12 UpperCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7809B7ED792669F3">39.17-13 PCentralSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X82A34BD681F24A94">39.17-14 JenningsSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7C08A8B77EC09CFF">39.17-15 DimensionsLoewyFactors</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X84112774812180DD">39.17-16 AscendingChain</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7C5029EE86D7FC96">39.17-17 IntermediateGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X781661FB78DC83B5">39.17-18 IntermediateSubgroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X80FC390C7F38A13F">39.18-1 NaturalHomomorphismByNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7E6EED0185B27C48">39.18-2 FactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7816FA867BF1B8ED">39.18-3 CommutatorFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7BB93B9778C5A0B2">39.18-4 MaximalAbelianQuotient</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7FC83E4C783572E7">39.18-5 HasAbelianFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7FAC018680B766B7">39.18-6 HasElementaryAbelianFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X822A3AB27919BC1E">39.18-7 CentralizerModulo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7DDE67C67E871336">39.19-1 ConjugacyClassSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7C5BBF487977B8CD">39.19-2 IsConjugacyClassSubgroupsRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7E986BF48393113A">39.19-3 ConjugacyClassesSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8486C25380853F9B">39.19-4 ConjugacyClassesMaximalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X798BF55C837DB188">39.19-5 MaximalSubgroupClassReps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X85DAFB7582A88463">39.19-6 LowIndexSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X80399CD4870FFC4B">39.19-7 AllSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X861CD8DA790D81C2">39.19-8 MaximalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X80237A847E24E6CF">39.19-9 NormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X82ECAA427C987318">39.19-10 MaximalNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X86FDD9BA819F5644">39.19-11 MinimalNormalSubgroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7B104E2C86166188">39.20-1 LatticeSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X78928A3582882BFD">39.20-2 ClassElementLattice</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7E5DF287825EE7BA">39.20-3 DotFileLatticeSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X815CDA447C5DB285">39.20-4 MaximalSubgroupsLattice</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8138997C871EDF96">39.20-5 MinimalSupergroupsLattice</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X87BE970D7B18E2C5">39.20-6 LowLayerSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X87FABD5F87AD2568">39.20-7 ContainedConjugates</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X79C3619C849F97B8">39.20-8 ContainingConjugates</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8111F50C798B0D76">39.20-9 MinimalFaithfulPermutationDegree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7BA3484E7AE0A0E1">39.20-10 RepresentativesPerfectSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7B2233D180DF77A1">39.20-11 ConjugacyClassesPerfectSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7BFE573187B4BEF8">39.20-12 Zuppos</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X82C12E2C81963B23">39.20-13 InfoLattice</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X86462A567DDBA6BC">39.21-1 LatticeByCyclicExtension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X78918D83835A0EDF">39.21-2 InvariantSubgroupsElementaryAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7AD7804A803910AC">39.21-3 SubgroupsSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7F60BBB8874DFE40">39.21-4 SizeConsiderFunction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X833C51BD7E7812C4">39.21-5 ExactSizeConsiderFunction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A2C774B7CFF3E07">39.21-6 InfoPcSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X82FD78AF7F80A0E2">39.22-1 GeneratorsSmallest</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7A258CCF79552198">39.22-2 LargestElementGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X81D15723804771E2">39.22-3 MinimalGeneratingSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X814DBABC878D5232">39.22-4 SmallGeneratingSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7D1574457B152333">39.22-5 IndependentGeneratorsOfAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X86F835DA8264A0CE">39.22-6 IndependentGeneratorExponents</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X847BEC137A49BAF4">39.23-1 <span class="Heading">OneCocycles</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7E6438D5834ACCDA">39.23-2 OneCoboundaries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X80400ABD7F40FAA0">39.23-3 OCOneCocycles</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X811E1CF07DABE924">39.23-4 ComplementClassesRepresentativesEA</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8199B1D27D487897">39.23-5 InfoCoh</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7F619DDA7DD6C43B">39.24-1 EpimorphismSchurCover</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7DD1E37987612042">39.24-2 SchurCover</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X792BC39D7CEB1D27">39.24-3 AbelianInvariantsMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X819E8AEC835F8CD1">39.24-4 Epicentre</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8739CD4686301A0E">39.24-5 NonabelianExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7E1C8CD77CDB9F71">39.24-6 EpimorphismNonabelianExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7BF8DB3D8300BB3F">39.24-7 IsCentralFactor</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.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_mj.html#X7DDA6BC1824F78FD">39.24-9 BasicSpinRepresentationOfSymmetricGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X844CFFDE80F6AD15">39.24-10 SchurCoverOfSymmetricGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7E0F4896795E34FC">39.24-11 DoubleCoverOfAlternatingGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.html#X865722987E0E19B6">39.25 <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_mj.html#X798F13EA810FB215">39.25-1 CanEasilyTestMembership</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7C2A89607BDFD920">39.25-2 CanEasilyComputeWithIndependentGensAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X83245C82835D496C">39.25-3 CanComputeSize</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X8268965487364912">39.25-4 CanComputeSizeAnySubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X82DDE00D82A32083">39.25-5 CanComputeIndex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X7BE7C36B84C23511">39.25-6 CanComputeIsSubset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap39_mj.html#X87D62C2C7C375E2D">39.25-7 KnowsHowToDecompose</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39_mj.html#X83A9997586694DC0">39.26 <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_mj.html#X84ABCA997D294B36">39.26-1 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">\(\{ 1, \ldots, 4 \}\)</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_mj.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_mj.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_mj.html#X84F59A2687C62763"><span class="RefLink">31.10-10</span></a>).</p>
<p>The operation <code class="func">Comm</code> (<a href="chap31_mj.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_mj.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_mj.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_mj.html#X8215999E835290F0"><span class="RefLink">37.2-1</span></a>).</p>
<p><a id="X7D8E473384DE9CD4" name="X7D8E473384DE9CD4"></a></p>
<h5>39.2-1 Group</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_mj.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_mj.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_mj.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_mj.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_mj.html#X7D8E473384DE9CD4"><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_mj.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_mj.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_mj.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_mj.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_mj.html#X82CBFF648574B830"><span class="RefLink">35.1-4</span></a>)) and associative (see <code class="func">IsAssociative</code> (<a href="chap35_mj.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_mj.html#X7A0747F17B50D967"><span class="RefLink">39.2-5</span></a>) if you want to perform such a test. (See <a href="chap13_mj.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_mj.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_mj.html#X7CBDD36E7B7BE286"><span class="RefLink">31.7</span></a> and <a href="chap31_mj.html#X7B58FDEF80338DD6"><span class="RefLink">31.8</span></a>. More functions that construct certain subgroups can be found in the sections <a href="chap39_mj.html#X804F0F037F06E25E"><span class="RefLink">39.11</span></a>, <a href="chap39_mj.html#X7C39EE3E836D6BC6"><span class="RefLink">39.12</span></a>, <a href="chap39_mj.html#X7FF0BBDD80E8F6BF"><span class="RefLink">39.13</span></a>, and <a href="chap39_mj.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 "universal" 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_mj.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_mj.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">\([\textit{G}:\textit{U}] = |\textit{G}| / |\textit{U}|\)</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_mj.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_mj.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">\(g \in \textit{G}\)</span> and <span class="SimpleMath">\(u \in \textit{U}\)</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_mj.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_mj.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 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">ClosureGroup(u,g.2);</span>
Group([ f2, f3, f4 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">ClosureGroup(u,[g.1,g.2]);</span>
Group([ f1, f2, f3, f4 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">ClosureGroup(u,Group(g.2*g.1));</span>
Group([ f1*f2^2, f3, f4 ])
</pre></div>
<p><a id="X81A20A397C308483" name="X81A20A397C308483"></a></p>
<h5>39.4-2 ClosureGroupAddElm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClosureGroupAddElm</code>( <var class="Arg">G</var>, <var class="Arg">elm</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">‣ ClosureGroupCompare</code>( <var class="Arg">G</var>, <var class="Arg">elm</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">‣ ClosureGroupIntest</code>( <var class="Arg">G</var>, <var class="Arg">elm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>These three functions together with <code class="func">ClosureGroupDefault</code> (<a href="chap39_mj.html#X82F59F6680D1B0D5"><span class="RefLink">39.4-3</span></a>) implement the main methods for <code class="func">ClosureGroup</code> (<a href="chap39_mj.html#X7D13FC1F8576FFD8"><span class="RefLink">39.4-1</span></a>). In the ordering given, they just add <var class="Arg">elm</var> to the generators, remove duplicates and identity elements, and test whether <var class="Arg">elm</var> is already contained in <var class="Arg">G</var>.</p>
<p><a id="X82F59F6680D1B0D5" name="X82F59F6680D1B0D5"></a></p>
<h5>39.4-3 ClosureGroupDefault</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClosureGroupDefault</code>( <var class="Arg">G</var>, <var class="Arg">elm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This functions returns the closure of the group <var class="Arg">G</var> with the element <var class="Arg">elm</var>. If <var class="Arg">G</var> has the attribute <code class="func">AsSSortedList</code> (<a href="chap30_mj.html#X856D927378C33548"><span class="RefLink">30.3-10</span></a>) then also the result has this attribute. This is used to implement the default method for <code class="func">Enumerator</code> (<a href="chap30_mj.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) and <code class="func">EnumeratorSorted</code> (<a href="chap30_mj.html#X80CD7DDC7D0C60D5"><span class="RefLink">30.3-3</span></a>).</p>
<p><a id="X7A7AF14A8052F055" name="X7A7AF14A8052F055"></a></p>
<h5>39.4-4 ClosureSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClosureSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">obj</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">‣ ClosureSubgroupNC</code>( <var class="Arg">G</var>, <var class="Arg">obj</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a group <var class="Arg">G</var> that stores a parent group (see <a href="chap31_mj.html#X7CBDD36E7B7BE286"><span class="RefLink">31.7</span></a>), <code class="func">ClosureSubgroup</code> calls <code class="func">ClosureGroup</code> (<a href="chap39_mj.html#X7D13FC1F8576FFD8"><span class="RefLink">39.4-1</span></a>) with the same arguments; if the result is a subgroup of the parent of <var class="Arg">G</var> then the parent of <var class="Arg">G</var> is set as parent of the result, otherwise an error is raised. The check whether the result is contained in the parent of <var class="Arg">G</var> is omitted by the <code class="code">NC</code> version. As a wrong parent might imply wrong properties this version should be used with care.</p>
<p><a id="X7E19F92284F6684E" name="X7E19F92284F6684E"></a></p>
<h4>39.5 <span class="Heading">Expressing Group Elements as Words in Generators</span></h4>
<p>Using homomorphisms (see chapter <a href="chap40_mj.html#X83702FC27B3C3098"><span class="RefLink">40</span></a>) is is possible to express group elements as words in given generators: Create a free group (see <code class="func">FreeGroup</code> (<a href="chap37_mj.html#X8215999E835290F0"><span class="RefLink">37.2-1</span></a>)) on the correct number of generators and create a homomorphism from this free group onto the group <var class="Arg">G</var> in whose generators you want to factorize. Then the preimage of an element of <var class="Arg">G</var> is a word in the free generators, that will map on this element again.</p>
<p><a id="X7FE8A3B08458A1BF" name="X7FE8A3B08458A1BF"></a></p>
<h5>39.5-1 EpimorphismFromFreeGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EpimorphismFromFreeGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a group <var class="Arg">G</var> with a known generating set, this attribute returns a homomorphism from a free group that maps the free generators to the groups generators.</p>
<p>The option <code class="code">names</code> can be used to prescribe a (print) name for the free generators.</p>
<p>The following example shows how to decompose elements of <span class="SimpleMath">\(S_4\)</span> in the generators <code class="code">(1,2,3,4)</code> and <code class="code">(1,2)</code>:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));</span>
Group([ (1,2,3,4), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=EpimorphismFromFreeGroup(g:names:=["x","y"]);</span>
[ x, y ] -> [ (1,2,3,4), (1,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesRepresentative(hom,(1,4));</span>
y^-1*x^-1*(x^-1*y^-1)^2*x
</pre></div>
<p>The following example stems from a real request to the <strong class="pkg">GAP</strong> Forum. In September 2000 a <strong class="pkg">GAP</strong> user working with puzzles wanted to express the permutation <code class="code">(1,2)</code> as a word as short as possible in particular generators of the symmetric group <span class="SimpleMath">\(S_{16}\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">perms := [ (1,2,3,7,11,10,9,5), (2,3,4,8,12,11,10,6),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (5,6,7,11,15,14,13,9), (6,7,8,12,16,15,14,10) ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">puzzle := Group( perms );;Size( puzzle );</span>
20922789888000
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=EpimorphismFromFreeGroup(puzzle:names:=["a", "b", "c", "d"]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">word := PreImagesRepresentative( hom, (1,2) );</span>
a^-1*c*b*c^-1*a*b^-1*a^-2*c^-1*a*b^-1*c*b
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( word );</span>
13
</pre></div>
<p><a id="X8357294D7B164106" name="X8357294D7B164106"></a></p>
<h5>39.5-2 Factorization</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Factorization</code>( <var class="Arg">G</var>, <var class="Arg">elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a factorization of <var class="Arg">elm</var> as word in the generators of the group <var class="Arg">G</var> given in the attribute <code class="func">GeneratorsOfGroup</code> (<a href="chap39_mj.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>). The attribute <code class="func">EpimorphismFromFreeGroup</code> (<a href="chap39_mj.html#X7FE8A3B08458A1BF"><span class="RefLink">39.5-1</span></a>) of <var class="Arg">G</var> will contain a map from the group <var class="Arg">G</var> to the free group in which the word is expressed. The attribute <code class="func">MappingGeneratorsImages</code> (<a href="chap40_mj.html#X863805187A24B5E3"><span class="RefLink">40.10-2</span></a>) of this map gives a list of generators and corresponding letters.</p>
<p>The algorithm used forms all elements of the group to ensure a short word is found. Therefore this function should <em>not</em> be used when the group <var class="Arg">G</var> has more than a few million elements. Because of this, one should not call this function within algorithms, but use homomorphisms instead.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SymmetricGroup( 6 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">r:=(3,4);; s:=(1,2,3,4,5,6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># create subgroup to force the system to use the generators r and s:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:= Subgroup(G, [ r, s ] );</span>
Group([ (3,4), (1,2,3,4,5,6) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Factorization( H, (1,2,3) );</span>
(x2*x1)^2*x2^-2
<span class="GAPprompt">gap></span> <span class="GAPinput">s*r*s*r*s^-2;</span>
(1,2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">MappingGeneratorsImages(EpimorphismFromFreeGroup(H));</span>
[ [ x1, x2 ], [ (3,4), (1,2,3,4,5,6) ] ]
</pre></div>
<p><a id="X871508DD808EB487" name="X871508DD808EB487"></a></p>
<h5>39.5-3 GrowthFunctionOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GrowthFunctionOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GrowthFunctionOfGroup</code>( <var class="Arg">G</var>, <var class="Arg">radius</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a group <var class="Arg">G</var> with a generating set given in <code class="func">GeneratorsOfGroup</code> (<a href="chap39_mj.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>), this function calculates the number of elements whose shortest expression as words in the generating set is of a particular length. It returns a list <var class="Arg">L</var>, whose <span class="SimpleMath">\(i+1\)</span> entry counts the number of elements whose shortest word expression has length <span class="SimpleMath">\(i\)</span>. If a maximal length <var class="Arg">radius</var> is given, only words up to length <var class="Arg">radius</var> are counted. Otherwise the group must be finite and all elements are enumerated.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GrowthFunctionOfGroup(MathieuGroup(12)); </span>
[ 1, 5, 19, 70, 255, 903, 3134, 9870, 25511, 38532, 16358, 382 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GrowthFunctionOfGroup(MathieuGroup(12),2);</span>
[ 1, 5, 19 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GrowthFunctionOfGroup(MathieuGroup(12),99);</span>
[ 1, 5, 19, 70, 255, 903, 3134, 9870, 25511, 38532, 16358, 382 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">free:=FreeGroup("a","b");</span>
<free group on the generators [ a, b ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">product:=free/ParseRelators(free,"a2,b3");</span>
<fp group on the generators [ a, b ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetIsFinite(product,false);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GrowthFunctionOfGroup(product,10);</span>
[ 1, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64 ]
</pre></div>
<p><a id="X87BF1B887C91CA2E" name="X87BF1B887C91CA2E"></a></p>
<h4>39.6 <span class="Heading">Structure Descriptions</span></h4>
<p><a id="X8199B74B84446971" name="X8199B74B84446971"></a></p>
<h5>39.6-1 StructureDescription</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StructureDescription</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The method for <code class="func">StructureDescription</code> exhibits a structure of the given group <var class="Arg">G</var> to some extent, using the strategy outlined below. The idea is to return a possibly short string which gives some insight in the structure of the considered group. It is intended primarily for small groups (order less than 100) or groups with few normal subgroups, in other cases, in particular large <span class="SimpleMath">\(p\)</span>-groups, it can be very costly. Furthermore, the string returned is -- as the action on chief factors is not described -- often not the most useful way to describe a group.</p>
<p>The string returned by <code class="func">StructureDescription</code> is <strong class="button">not</strong> an isomorphism invariant: non-isomorphic groups can have the same string value, and two isomorphic groups in different representations can produce different strings. The value returned by <code class="func">StructureDescription</code> is a string of the following form:</p>
<div class="example"><pre>
StructureDescription(<G>) ::=
1 ; trivial group
| C<size> ; cyclic group
| A<degree> ; alternating group
| S<degree> ; symmetric group
| D<size> ; dihedral group
| Q<size> ; quaternion group
| QD<size> ; quasidihedral group
| PSL(<n>,<q>) ; projective special linear group
| SL(<n>,<q>) ; special linear group
| GL(<n>,<q>) ; general linear group
| PSU(<n>,<q>) ; proj. special unitary group
| O(2<n>+1,<q>) ; orthogonal group, type B
| O+(2<n>,<q>) ; orthogonal group, type D
| O-(2<n>,<q>) ; orthogonal group, type 2D
| PSp(2<n>,<q>) ; proj. special symplectic group
| Sz(<q>) ; Suzuki group
| Ree(<q>) ; Ree group (type 2F or 2G)
| E(6,<q>) | E(7,<q>) | E(8,<q>) ; Lie group of exceptional type
| 2E(6,<q>) | F(4,<q>) | G(2,<q>)
| 3D(4,<q>) ; Steinberg triality group
| M11 | M12 | M22 | M23 | M24
| J1 | J2 | J3 | J4 | Co1 | Co2
| Co3 | Fi22 | Fi23 | Fi24' | Suz
| HS | McL | He | HN | Th | B
| M | ON | Ly | Ru ; sporadic simple group
| 2F(4,2)' ; Tits group
| PerfectGroup(<size>,<id>) ; the indicated group from the
; library of perfect groups
| A x B ; direct product
| N : H ; semidirect product
| C(G) . G/C(G) = G' . G/G' ; non-split extension
; (equal alternatives and
; trivial extensions omitted)
| Phi(G) . G/Phi(G) ; non-split extension:
; Frattini subgroup and
; Frattini factor group
</pre></div>
<p>Note that the <code class="func">StructureDescription</code> is only <em>one</em> possible way of building up the given group from smaller pieces.</p>
<p>The option "short" is recognized - if this option is set, an abbreviated output format is used (e.g. <code class="code">"6x3"</code> instead of <code class="code">"C6 x C3"</code>).</p>
<p>If the <code class="func">Name</code> (<a href="chap12_mj.html#X7F14EF9D81432113"><span class="RefLink">12.8-2</span></a>) attribute is not bound, but <code class="func">StructureDescription</code> is, <code class="func">View</code> (<a href="chap6_mj.html#X851902C583B84CDC"><span class="RefLink">6.3-3</span></a>) prints the value of the attribute <code class="func">StructureDescription</code>. The <code class="func">Print</code> (<a href="chap6_mj.html#X7AFA64D97A1F39A3"><span class="RefLink">6.3-4</span></a>)ed representation of a group is not affected by computing a <code class="func">StructureDescription</code>.</p>
<p>The strategy used to compute a <code class="func">StructureDescription</code> is as follows:</p>
<dl>
<dt><strong class="Mark">1.</strong></dt>
<dd><p>Lookup in a precomputed list, if the order of <var class="Arg">G</var> is not larger than 100 and not equal to 64 or 96.</p>
</dd>
<dt><strong class="Mark">2.</strong></dt>
<dd><p>If <var class="Arg">G</var> is abelian, then decompose it into cyclic factors in "elementary divisors style". For example, <code class="code">"C2 x C3 x C3"</code> is <code class="code">"C6 x C3"</code>. For infinite abelian groups, <code class="code">"C0"</code> denotes the group of integers.</p>
</dd>
<dt><strong class="Mark">3.</strong></dt>
<dd><p>Recognize alternating groups, symmetric groups, dihedral groups, quasidihedral groups, quaternion groups, PSL's, SL's, GL's and simple groups not listed so far as basic building blocks.</p>
</dd>
<dt><strong class="Mark">4.</strong></dt>
<dd><p>Decompose <var class="Arg">G</var> into a direct product of irreducible factors.</p>
</dd>
<dt><strong class="Mark">5.</strong></dt>
<dd><p>Recognize semidirect products <var class="Arg">G</var>=<span class="SimpleMath">\(N\)</span>:<span class="SimpleMath">\(H\)</span>, where <span class="SimpleMath">\(N\)</span> is normal. Select a pair <span class="SimpleMath">\(N\)</span>, <span class="SimpleMath">\(H\)</span> with the following preferences:</p>
<dl>
<dt><strong class="Mark">1.</strong></dt>
<dd><p>if <var class="Arg">G</var> is defined as a semidirect product of <span class="SimpleMath">\(N\)</span>, <span class="SimpleMath">\(H\)</span> then select <span class="SimpleMath">\(N\)</span>, <span class="SimpleMath">\(H\)</span>,</p>
</dd>
<dt><strong class="Mark">2.</strong></dt>
<dd><p>if <var class="Arg">G</var> is solvable, then select a solvable normal Hall subgroup <span class="SimpleMath">\(N\)</span>, if exists, and consider the semidirect decomposition of <span class="SimpleMath">\(N\)</span> and <span class="SimpleMath">\(\textit{G}/N\)</span>,</p>
</dd>
<dt><strong class="Mark">3.</strong></dt>
<dd><p>find any nontrivial normal subgroup <span class="SimpleMath">\(N\)</span> which has a complement <span class="SimpleMath">\(H\)</span>.</p>
</dd>
</dl>
<p>The option "nice" is recognized. If this option is set, then all semidirect products are computed in order to find a possibly nicer presentation. Note, that this may take a very long time if <var class="Arg">G</var> has many normal subgroups, e.g. if <span class="SimpleMath">\(\textit{G}/\textit{G}'\)</span> has many cyclic factors. If the option "nice" is set, then GAP would select a pair <span class="SimpleMath">\(N\)</span>, <span class="SimpleMath">\(H\)</span> with the following preferences:</p>
<dl>
<dt><strong class="Mark">1.</strong></dt>
<dd><p><span class="SimpleMath">\(H\)</span> is abelian</p>
</dd>
<dt><strong class="Mark">2.</strong></dt>
<dd><p><span class="SimpleMath">\(N\)</span> is abelian</p>
</dd>
<dt><strong class="Mark">2a.</strong></dt>
<dd><p><span class="SimpleMath">\(N\)</span> has many abelian invariants</p>
</dd>
<dt><strong class="Mark">3.</strong></dt>
<dd><p><span class="SimpleMath">\(N\)</span> is a direct product</p>
</dd>
<dt><strong class="Mark">3a.</strong></dt>
<dd><p><span class="SimpleMath">\(N\)</span> has many direct factors</p>
</dd>
<dt><strong class="Mark">4.</strong></dt>
<dd><p><span class="SimpleMath">\(\phi: H \rightarrow\)</span> Aut(<span class="SimpleMath">\(N\)</span>), <span class="SimpleMath">\(h \mapsto (n \mapsto n^h)\)</span> is injective.</p>
</dd>
</dl>
</dd>
<dt><strong class="Mark">6.</strong></dt>
<dd><p>Fall back to non-splitting extensions: If the centre or the commutator factor group is non-trivial, write <var class="Arg">G</var> as <span class="SimpleMath">\(Z(\textit{G})\)</span>.<span class="SimpleMath">\(\textit{G}/Z(\textit{G})\)</span> or <span class="SimpleMath">\(\textit{G}'\)</span>.<span class="SimpleMath">\(\textit{G}/\textit{G}'\)</span>, respectively. Otherwise if the Frattini subgroup is non-trivial, write <var class="Arg">G</var> as <span class="SimpleMath">\(\Phi\)</span>(<var class="Arg">G</var>).<var class="Arg">G</var>/<span class="SimpleMath">\(\Phi\)</span>(<var class="Arg">G</var>).</p>
</dd>
<dt><strong class="Mark">7.</strong></dt>
<dd><p>If no decomposition is found (maybe this is not the case for any finite group), try to identify <var class="Arg">G</var> in the perfect groups library. If this fails also, then return a string describing this situation.</p>
</dd>
</dl>
<p>Note that <code class="func">StructureDescription</code> is <em>not</em> intended to be a research tool, but rather an educational tool. The reasons for this are as follows:</p>
<dl>
<dt><strong class="Mark">1.</strong></dt>
<dd><p>"Most" groups do not have "nice" decompositions. This is in some contrast to what is often taught in elementary courses on group theory, where it is sometimes suggested that basically every group can be written as iterated direct or semidirect product of cyclic groups and nonabelian simple groups.</p>
</dd>
<dt><strong class="Mark">2.</strong></dt>
<dd><p>In particular many <span class="SimpleMath">\(p\)</span>-groups have very "similar" structure, and <code class="func">StructureDescription</code> can only exhibit a little of it. Changing this would likely make the output not essentially easier to read than a pc presentation.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := AllSmallGroups(12);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(l,StructureDescription);; l;</span>
[ C3 : C4, C12, A4, D12, C6 x C2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallGroups(40),G->StructureDescription(G:short));</span>
[ "5:8", "40", "5:8", "5:Q8", "4xD10", "D40", "2x(5:4)", "(10x2):2",
"20x2", "5xD8", "5xQ8", "2x(5:4)", "2^2xD10", "10x2^2" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllTransitiveGroups(DegreeAction,6),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> G->StructureDescription(G:short));</span>
[ "6", "S3", "D12", "A4", "3xS3", "2xA4", "S4", "S4", "S3xS3",
"(3^2):4", "2xS4", "A5", "(S3xS3):2", "S5", "A6", "S6" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(SmallGroup(504,7));</span>
"C7 : (C9 x Q8)"
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(SmallGroup(504,7):nice);</span>
"(C7 : Q8) : C9"
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(AbelianGroup([0,2,3]));</span>
"C0 x C6"
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(AbelianGroup([0,0,0,2,3,6]):short);</span>
"0^3x6^2"
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(PSL(4,2));</span>
"A8"
</pre></div>
<p><a id="X81002AA87DDBC02F" name="X81002AA87DDBC02F"></a></p>
<h4>39.7 <span class="Heading">Cosets</span></h4>
<p><a id="X8412ABD57986B9FC" name="X8412ABD57986B9FC"></a></p>
<h5>39.7-1 RightCoset</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightCoset</code>( <var class="Arg">U</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the right coset of <var class="Arg">U</var> with representative <var class="Arg">g</var>, which is the set of all elements of the form <span class="SimpleMath">\(ug\)</span> for all <span class="SimpleMath">\(u \in \textit{U}\)</span>. <var class="Arg">g</var> must be an element of a larger group <var class="Arg">G</var> which contains <var class="Arg">U</var>. For element operations such as <code class="keyw">in</code> a right coset behaves like a set of group elements.</p>
<p>Right cosets are external orbits for the action of <var class="Arg">U</var> which acts via <code class="func">OnLeftInverse</code> (<a href="chap41_mj.html#X832DF5327ECA0E44"><span class="RefLink">41.2-3</span></a>). Of course the action of a larger group <var class="Arg">G</var> on right cosets is via <code class="func">OnRight</code> (<a href="chap41_mj.html#X7960924D84B5B18F"><span class="RefLink">41.2-2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:=Group((1,2,3), (1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:=RightCoset(u,(2,3,4));</span>
RightCoset(Group( [ (1,2,3), (1,2) ] ),(2,3,4))
<span class="GAPprompt">gap></span> <span class="GAPinput">ActingDomain(c);</span>
Group([ (1,2,3), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Representative(c);</span>
(2,3,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(c);</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">AsList(c);</span>
[ (2,3,4), (1,4,2), (1,3,4,2), (1,3)(2,4), (2,4), (1,4,2,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBiCoset(c);</span>
false
</pre></div>
<p><a id="X835F48248571364F" name="X835F48248571364F"></a></p>
<h5>39.7-2 RightCosets</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightCosets</code>( <var class="Arg">G</var>, <var class="Arg">U</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">‣ RightCosetsNC</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes a duplicate free list of right cosets <var class="Arg">U</var> <span class="SimpleMath">\(g\)</span> for <span class="SimpleMath">\(g \in\)</span> <var class="Arg">G</var>. A set of representatives for the elements in this list forms a right transversal of <var class="Arg">U</var> in <var class="Arg">G</var>. (By inverting the representatives one obtains a list of representatives of the left cosets of <var class="Arg">U</var>.) The <code class="code">NC</code> version does not check whether <var class="Arg">U</var> is a subgroup of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RightCosets(g,u);</span>
[ RightCoset(Group( [ (1,2,3), (1,2) ] ),()),
RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,3)(2,4)),
RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,4)(2,3)),
RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,2)(3,4)) ]
</pre></div>
<p><a id="X85884F177B5D98AE" name="X85884F177B5D98AE"></a></p>
<h5>39.7-3 CanonicalRightCosetElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanonicalRightCosetElement</code>( <var class="Arg">U</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a "canonical" representative of the right coset <var class="Arg">U</var> <var class="Arg">g</var> which is independent of the given representative <var class="Arg">g</var>. This can be used to compare cosets by comparing their canonical representatives.</p>
<p>The representative chosen to be the "canonical" one is representation dependent and only guaranteed to remain the same within one <strong class="pkg">GAP</strong> session.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CanonicalRightCosetElement(u,(2,4,3));</span>
(3,4)
</pre></div>
<p><a id="X7D7625A1861D9DAB" name="X7D7625A1861D9DAB"></a></p>
<h5>39.7-4 IsRightCoset</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightCoset</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>The category of right cosets.</p>
<p><strong class="pkg">GAP</strong> does not provide left cosets as a separate data type, but as the left coset <span class="SimpleMath">\(gU\)</span> consists of exactly the inverses of the elements of the right coset <span class="SimpleMath">\(Ug^{{-1}}\)</span> calculations with left cosets can be emulated using right cosets by inverting the representatives.</p>
<p><a id="X78F4F0D8838F5ABF" name="X78F4F0D8838F5ABF"></a></p>
<h5>39.7-5 IsBiCoset</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBiCoset</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A (right) coset <span class="SimpleMath">\(Ug\)</span> is considered a <em>bicoset</em> if its set of elements simultaneously forms a left coset for the same subgroup. This is the case if and only if the coset representative <span class="SimpleMath">\(g\)</span> normalizes the subgroup <span class="SimpleMath">\(U\)</span>.</p>
<p><a id="X82F6ABE378B928D1" name="X82F6ABE378B928D1"></a></p>
<h5>39.7-6 CosetDecomposition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CosetDecomposition</code>( <var class="Arg">G</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a finite group <var class="Arg">G</var> and a subgroup <span class="SimpleMath">\(\textit{S}\le\textit{G}\)</span> this function returns a partition of the elements of <var class="Arg">G</var> according to the (right) cosets of <var class="Arg">S</var>. The result is a list of lists, each sublist corresponding to one coset. The first sublist is the elements list of the subgroup, the other lists are arranged accordingly.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CosetDecomposition(SymmetricGroup(4),SymmetricGroup(3)); </span>
[ [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ],
[ (1,4), (1,4)(2,3), (1,2,4), (1,2,3,4), (1,3,2,4), (1,3,4) ],
[ (1,4,2), (1,4,2,3), (2,4), (2,3,4), (1,3)(2,4), (1,3,4,2) ],
[ (1,4,3), (1,4,3,2), (1,2,4,3), (1,2)(3,4), (2,4,3), (3,4) ] ]
</pre></div>
<p><a id="X83C723878230D616" name="X83C723878230D616"></a></p>
<h4>39.8 <span class="Heading">Transversals</span></h4>
<p><a id="X85C65D06822E716F" name="X85C65D06822E716F"></a></p>
<h5>39.8-1 RightTransversal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightTransversal</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A right transversal <span class="SimpleMath">\(t\)</span> is a list of representatives for the set <span class="SimpleMath">\(\textit{U} \setminus \textit{G}\)</span> of right cosets (consisting of cosets <span class="SimpleMath">\(Ug\)</span>) of <span class="SimpleMath">\(U\)</span> in <span class="SimpleMath">\(G\)</span>.</p>
<p>The object returned by <code class="func">RightTransversal</code> is not a plain list, but an object that behaves like an immutable list of length <span class="SimpleMath">\([\textit{G}:\textit{U}]\)</span>, except if <var class="Arg">U</var> is the trivial subgroup of <var class="Arg">G</var> in which case <code class="func">RightTransversal</code> may return the sorted plain list of coset representatives.</p>
<p>The operation <code class="func">PositionCanonical</code> (<a href="chap21_mj.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>), called for a transversal <span class="SimpleMath">\(t\)</span> and an element <span class="SimpleMath">\(g\)</span> of <var class="Arg">G</var>, will return the position of the representative in <span class="SimpleMath">\(t\)</span> that lies in the same coset of <var class="Arg">U</var> as the element <span class="SimpleMath">\(g\)</span> does. (In comparison, <code class="func">Position</code> (<a href="chap21_mj.html#X79975EC6783B4293"><span class="RefLink">21.16-1</span></a>) will return <code class="keyw">fail</code> if the element is not equal to the representative.) Functions that implement group actions such as <code class="func">Action</code> (<a href="chap41_mj.html#X85A8E93D786C3C9C"><span class="RefLink">41.7-2</span></a>) or <code class="func">Permutation</code> (<a href="chap41_mj.html#X7807A33381DCAB26"><span class="RefLink">41.9-1</span></a>) (see Chapter <a href="chap41_mj.html#X87115591851FB7F4"><span class="RefLink">41</span></a>) use <code class="func">PositionCanonical</code> (<a href="chap21_mj.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>), therefore it is possible to "act" on a right transversal to implement the action on the cosets. This is often much more efficient than acting on cosets.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:=Subgroup(g,[(1,2,3),(1,2)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rt:=RightTransversal(g,u);</span>
RightTransversal(Group([ (1,2,3,4), (1,2) ]),Group([ (1,2,3), (1,2) ]))
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(rt);</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">Position(rt,(1,2,3));</span>
fail
</pre></div>
<p>Note that the elements of a right transversal are not necessarily "canonical" in the sense of <code class="func">CanonicalRightCosetElement</code> (<a href="chap39_mj.html#X85884F177B5D98AE"><span class="RefLink">39.7-3</span></a>), but we may compute a list of canonical coset representatives by calling that function. (See also <code class="func">PositionCanonical</code> (<a href="chap21_mj.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>).)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(RightTransversal(g,u),i->CanonicalRightCosetElement(u,i));</span>
[ (), (2,3,4), (1,2,3,4), (3,4) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PositionCanonical(rt,(1,2,3));</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">rt[1];</span>
()
</pre></div>
<p><a id="X78B98B257E981046" name="X78B98B257E981046"></a></p>
<h4>39.9 <span class="Heading">Double Cosets</span></h4>
<p><a id="X7E51ED757D17254B" name="X7E51ED757D17254B"></a></p>
<h5>39.9-1 DoubleCoset</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DoubleCoset</code>( <var class="Arg">U</var>, <var class="Arg">g</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The groups <var class="Arg">U</var> and <var class="Arg">V</var> must be subgroups of a common supergroup <var class="Arg">G</var> of which <var class="Arg">g</var> is an element. This command constructs the double coset <var class="Arg">U</var> <var class="Arg">g</var> <var class="Arg">V</var> which is the set of all elements of the form <span class="SimpleMath">\(ugv\)</span> for any <span class="SimpleMath">\(u \in \textit{U}\)</span>, <span class="SimpleMath">\(v \in \textit{V}\)</span>. For element operations such as <code class="keyw">in</code>, a double coset behaves like a set of group elements. The double coset stores <var class="Arg">U</var> in the attribute <code class="code">LeftActingGroup</code>, <var class="Arg">g</var> as <code class="func">Representative</code> (<a href="chap30_mj.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>), and <var class="Arg">V</var> as <code class="code">RightActingGroup</code>.</p>
<p><a id="X7F53DABD79BA4F72" name="X7F53DABD79BA4F72"></a></p>
<h5>39.9-2 RepresentativesContainedRightCosets</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesContainedRightCosets</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A double coset <span class="SimpleMath">\(\textit{D} = U g V\)</span> can be considered as a union of right cosets <span class="SimpleMath">\(U h_i\)</span>. (It is the union of the orbit of <span class="SimpleMath">\(U g\)</span> under right multiplication by <span class="SimpleMath">\(V\)</span>.) For a double coset <var class="Arg">D</var> this function returns a set of representatives <span class="SimpleMath">\(h_i\)</span> such that <var class="Arg">D</var> <span class="SimpleMath">\(= \bigcup_{{h_i}} U h_i\)</span>. The representatives returned are canonical for <span class="SimpleMath">\(U\)</span> (see <code class="func">CanonicalRightCosetElement</code> (<a href="chap39_mj.html#X85884F177B5D98AE"><span class="RefLink">39.7-3</span></a>)) and form a set.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:=Subgroup(g,[(1,2,3),(1,2)]);;v:=Subgroup(g,[(3,4)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:=DoubleCoset(u,(2,4),v);</span>
DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(2,4),Group( [ (3,4) ] ))
<span class="GAPprompt">gap></span> <span class="GAPinput">(1,2,3) in c;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">(2,3,4) in c;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">LeftActingGroup(c);</span>
Group([ (1,2,3), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">RightActingGroup(c);</span>
Group([ (3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativesContainedRightCosets(c);</span>
[ (2,3,4) ]
</pre></div>
<p><a id="X7A5EFABB86E6D4D5" name="X7A5EFABB86E6D4D5"></a></p>
<h5>39.9-3 DoubleCosets</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DoubleCosets</code>( <var class="Arg">G</var>, <var class="Arg">U</var>, <var class="Arg">V</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">‣ DoubleCosetsNC</code>( <var class="Arg">G</var>, <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes a duplicate free list of all double cosets <var class="Arg">U</var> <span class="SimpleMath">\(g\)</span> <var class="Arg">V</var> for <span class="SimpleMath">\(g \in \textit{G}\)</span>. The groups <var class="Arg">U</var> and <var class="Arg">V</var> must be subgroups of the group <var class="Arg">G</var>. The <code class="code">NC</code> version does not check whether <var class="Arg">U</var> and <var class="Arg">V</var> are subgroups of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">dc:=DoubleCosets(g,u,v);</span>
[ DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(),Group( [ (3,4) ] )),
DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(1,3)(2,4),Group(
[ (3,4) ] )), DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(1,4)
(2,3),Group( [ (3,4) ] )) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(dc,Representative);</span>
[ (), (1,3)(2,4), (1,4)(2,3) ]
</pre></div>
<p><a id="X85ED464F878EF24C" name="X85ED464F878EF24C"></a></p>
<h5>39.9-4 IsDoubleCoset</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDoubleCoset</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>The category of double cosets.</p>
<p><a id="X7A25B1C886CF8C6A" name="X7A25B1C886CF8C6A"></a></p>
<h5>39.9-5 DoubleCosetRepsAndSizes</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DoubleCosetRepsAndSizes</code>( <var class="Arg">G</var>, <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a list of double coset representatives and their sizes, the entries are lists of the form <span class="SimpleMath">\([ r, n ]\)</span> where <span class="SimpleMath">\(r\)</span> and <span class="SimpleMath">\(n\)</span> are an element of the double coset and the size of the coset, respectively. This operation is faster than <code class="func">DoubleCosetsNC</code> (<a href="chap39_mj.html#X7A5EFABB86E6D4D5"><span class="RefLink">39.9-3</span></a>) because no double coset objects have to be created.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">dc:=DoubleCosetRepsAndSizes(g,u,v);</span>
[ [ (), 12 ], [ (1,3)(2,4), 6 ], [ (1,4)(2,3), 6 ] ]
</pre></div>
<p><a id="X84AE7EE77E5FB30E" name="X84AE7EE77E5FB30E"></a></p>
<h5>39.9-6 InfoCoset</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoCoset</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>The information function for coset and double coset operations is <code class="func">InfoCoset</code>.</p>
<p><a id="X7D474F8F87E4E5D9" name="X7D474F8F87E4E5D9"></a></p>
<h4>39.10 <span class="Heading">Conjugacy Classes</span></h4>
<p><a id="X7B2F207F7F85F5B8" name="X7B2F207F7F85F5B8"></a></p>
<h5>39.10-1 ConjugacyClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClass</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>creates the conjugacy class in <var class="Arg">G</var> with representative <var class="Arg">g</var>. This class is an external set, so functions such as <code class="func">Representative</code> (<a href="chap30_mj.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>) (which returns <var class="Arg">g</var>), <code class="func">ActingDomain</code> (<a href="chap41_mj.html#X7B9DB15D80CE28B4"><span class="RefLink">41.12-3</span></a>) (which returns <var class="Arg">G</var>), <code class="func">StabilizerOfExternalSet</code> (<a href="chap41_mj.html#X7BAFF02B7D6DF9F2"><span class="RefLink">41.12-10</span></a>) (which returns the centralizer of <var class="Arg">g</var>) and <code class="func">AsList</code> (<a href="chap30_mj.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) work for it.</p>
<p>A conjugacy class is an external orbit (see <code class="func">ExternalOrbit</code> (<a href="chap41_mj.html#X7FB656AE7A066C35"><span class="RefLink">41.12-9</span></a>)) of group elements with the group acting by conjugation on it. Thus element tests or operation representatives can be computed. The attribute <code class="func">Centralizer</code> (<a href="chap35_mj.html#X7DE33AFC823C7873"><span class="RefLink">35.4-4</span></a>) gives the centralizer of the representative (which is the same result as <code class="func">StabilizerOfExternalSet</code> (<a href="chap41_mj.html#X7BAFF02B7D6DF9F2"><span class="RefLink">41.12-10</span></a>)). (This is a slight abuse of notation: This is <em>not</em> the centralizer of the class as a <em>set</em> which would be the standard behaviour of <code class="func">Centralizer</code> (<a href="chap35_mj.html#X7DE33AFC823C7873"><span class="RefLink">35.4-4</span></a>).)</p>
<p><a id="X871B570284BBA685" name="X871B570284BBA685"></a></p>
<h5>39.10-2 ConjugacyClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClasses</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the conjugacy classes of elements of <var class="Arg">G</var> as a list of class objects of <var class="Arg">G</var> (see <code class="func">ConjugacyClass</code> (<a href="chap39_mj.html#X7B2F207F7F85F5B8"><span class="RefLink">39.10-1</span></a>) for details). It is guaranteed that the class of the identity is in the first position, the further arrangement depends on the method chosen (and might be different for equal but not identical groups).</p>
<p>For very small groups (of size up to 500) the classes will be computed by the conjugation action of <var class="Arg">G</var> on itself (see <code class="func">ConjugacyClassesByOrbits</code> (<a href="chap39_mj.html#X852B3634789D770E"><span class="RefLink">39.10-4</span></a>)). This can be deliberately switched off using the "<code class="code">noaction</code>" option shown below.</p>
<p>For solvable groups, the default method to compute the classes is by homomorphic lift (see section <a href="chap45_mj.html#X79DCCF6D80351859"><span class="RefLink">45.17</span></a>).</p>
<p>For other groups the method of <a href="chapBib_mj.html#biBHulpkeClasses">[Hul00]</a> is employed.</p>
<p><code class="func">ConjugacyClasses</code> supports the following options that can be used to modify this strategy:</p>
<dl>
<dt><strong class="Mark"><code class="code">random</code></strong></dt>
<dd><p>The classes are computed by random search. See <code class="func">ConjugacyClassesByRandomSearch</code> (<a href="chap39_mj.html#X7D6ED84C86C2979B"><span class="RefLink">39.10-3</span></a>) below.</p>
</dd>
<dt><strong class="Mark"><code class="code">action</code></strong></dt>
<dd><p>The classes are computed by action of <var class="Arg">G</var> on itself. See <code class="func">ConjugacyClassesByOrbits</code> (<a href="chap39_mj.html#X852B3634789D770E"><span class="RefLink">39.10-4</span></a>) below.</p>
</dd>
<dt><strong class="Mark"><code class="code">noaction</code></strong></dt>
<dd><p>Even for small groups <code class="func">ConjugacyClassesByOrbits</code> (<a href="chap39_mj.html#X852B3634789D770E"><span class="RefLink">39.10-4</span></a>) is not used as a default. This can be useful if the elements of the group use a lot of memory.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cl:=ConjugacyClasses(g);</span>
[ ()^G, (1,2)^G, (1,2)(3,4)^G, (1,2,3)^G, (1,2,3,4)^G ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Representative(cl[3]);Centralizer(cl[3]);</span>
(1,2)(3,4)
Group([ (1,2), (1,3)(2,4), (3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Centralizer(cl[5]));</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(cl[2]);</span>
6
</pre></div>
<p>In general, you will not need to have to influence the method, but simply call <code class="func">ConjugacyClasses</code> –<strong class="pkg">GAP</strong> will try to select a suitable method on its own. The method specifications are provided here mainly for expert use.</p>
<p><a id="X7D6ED84C86C2979B" name="X7D6ED84C86C2979B"></a></p>
<h5>39.10-3 ConjugacyClassesByRandomSearch</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClassesByRandomSearch</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the classes of the group <var class="Arg">G</var> by random search. This works very efficiently for almost simple groups.</p>
<p>This function is also accessible via the option <code class="code">random</code> to the function <code class="func">ConjugacyClass</code> (<a href="chap39_mj.html#X7B2F207F7F85F5B8"><span class="RefLink">39.10-1</span></a>).</p>
<p><a id="X852B3634789D770E" name="X852B3634789D770E"></a></p>
<h5>39.10-4 ConjugacyClassesByOrbits</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClassesByOrbits</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the classes of the group <var class="Arg">G</var> as orbits of <var class="Arg">G</var> on its elements. This can be quick but unsurprisingly may also take a lot of memory if <var class="Arg">G</var> becomes larger. All the classes will store their element list and thus a membership test will be quick as well.</p>
<p>This function is also accessible via the option <code class="code">action</code> to the function <code class="func">ConjugacyClass</code> (<a href="chap39_mj.html#X7B2F207F7F85F5B8"><span class="RefLink">39.10-1</span></a>).</p>
<p>Typically, for small groups (roughly of order up to <span class="SimpleMath">\(10^3\)</span>) the computation of classes as orbits under the action is fastest; memory restrictions (and the increasing cost of eliminating duplicates) make this less efficient for larger groups.</p>
<p>Calculation by random search has the smallest memory requirement, but in generally performs worse, the more classes are there.</p>
<p>The following example shows the effect of this for a small group with many classes:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Group((4,5)(6,7,8),(1,2,3)(5,6,9));;ConjugacyClasses(h:noaction);;time;</span>
110
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Group((4,5)(6,7,8),(1,2,3)(5,6,9));;ConjugacyClasses(h:random);;time;</span>
300
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Group((4,5)(6,7,8),(1,2,3)(5,6,9));;ConjugacyClasses(h:action);;time;</span>
30
</pre></div>
<p><a id="X8733F87B7E4C9903" name="X8733F87B7E4C9903"></a></p>
<h5>39.10-5 NrConjugacyClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClasses</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the number of conjugacy classes of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClasses(g);</span>
5
</pre></div>
<p><a id="X7BD2A4427B7FE248" name="X7BD2A4427B7FE248"></a></p>
<h5>39.10-6 RationalClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RationalClass</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>creates the rational class in <var class="Arg">G</var> with representative <var class="Arg">g</var>. A rational class consists of all elements that are conjugate to <var class="Arg">g</var> or to an <span class="SimpleMath">\(i\)</span>-th power of <var class="Arg">g</var> where <span class="SimpleMath">\(i\)</span> is coprime to the order of <span class="SimpleMath">\(g\)</span>. Thus a rational class can be interpreted as a conjugacy class of cyclic subgroups. A rational class is an external set (<code class="func">IsExternalSet</code> (<a href="chap41_mj.html#X8264C3C479FF0A8B"><span class="RefLink">41.12-1</span></a>)) of group elements with the group acting by conjugation on it, but not an external orbit.</p>
<p><a id="X81E9EF0A811072E8" name="X81E9EF0A811072E8"></a></p>
<h5>39.10-7 RationalClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RationalClasses</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the rational classes of the group <var class="Arg">G</var>. (See <code class="func">RationalClass</code> (<a href="chap39_mj.html#X7BD2A4427B7FE248"><span class="RefLink">39.10-6</span></a>).)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RationalClasses(DerivedSubgroup(g));</span>
[ RationalClass( AlternatingGroup( [ 1 .. 4 ] ), () ),
RationalClass( AlternatingGroup( [ 1 .. 4 ] ), (1,2)(3,4) ),
RationalClass( AlternatingGroup( [ 1 .. 4 ] ), (1,2,3) ) ]
</pre></div>
<p><a id="X877691247DE23386" name="X877691247DE23386"></a></p>
<h5>39.10-8 GaloisGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GaloisGroup</code>( <var class="Arg">ratcl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Suppose that <var class="Arg">ratcl</var> is a rational class of a group <span class="SimpleMath">\(G\)</span> with representative <span class="SimpleMath">\(g\)</span>. The exponents <span class="SimpleMath">\(i\)</span> for which <span class="SimpleMath">\(g^i\)</span> lies already in the ordinary conjugacy class of <span class="SimpleMath">\(g\)</span>, form a subgroup of the <em>prime residue class group</em> <span class="SimpleMath">\(P_n\)</span> (see <code class="func">PrimitiveRootMod</code> (<a href="chap15_mj.html#X82440BB9812FF148"><span class="RefLink">15.3-3</span></a>)), the so-called <em>Galois group</em> of the rational class. The prime residue class group <span class="SimpleMath">\(P_n\)</span> is obtained in <strong class="pkg">GAP</strong> as <code class="code">Units( Integers mod <var class="Arg">n</var> )</code>, the unit group of a residue class ring. The Galois group of a rational class <var class="Arg">ratcl</var> is stored in the attribute <code class="func">GaloisGroup</code> as a subgroup of this group.</p>
<p><a id="X83DD148D7DA2ABA9" name="X83DD148D7DA2ABA9"></a></p>
<h5>39.10-9 <span class="Heading">IsConjugate</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsConjugate</code>( <var class="Arg">G</var>, <var class="Arg">x</var>, <var class="Arg">y</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">‣ IsConjugate</code>( <var class="Arg">G</var>, <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>tests whether the elements <var class="Arg">x</var> and <var class="Arg">y</var> or the subgroups <var class="Arg">U</var> and <var class="Arg">V</var> are conjugate under the action of <var class="Arg">G</var>. (They do not need to be <em>contained in</em> <var class="Arg">G</var>.) This command is only a shortcut to <code class="func">RepresentativeAction</code> (<a href="chap41_mj.html#X857DC7B085EB0539"><span class="RefLink">41.6-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsConjugate(g,Group((1,2,3,4),(1,3)),Group((1,3,2,4),(1,2)));</span>
true
</pre></div>
<p><code class="func">RepresentativeAction</code> (<a href="chap41_mj.html#X857DC7B085EB0539"><span class="RefLink">41.6-1</span></a>) can be used to obtain conjugating elements.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeAction(g,(1,2),(3,4));</span>
(1,3)(2,4)
</pre></div>
<p><a id="X81A92F828400FC8A" name="X81A92F828400FC8A"></a></p>
<h5>39.10-10 NthRootsInGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NthRootsInGroup</code>( <var class="Arg">G</var>, <var class="Arg">e</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">e</var> be an element in the group <var class="Arg">G</var>. This function returns a list of all those elements in <var class="Arg">G</var> whose <var class="Arg">n</var>-th power is <var class="Arg">e</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NthRootsInGroup(g,(1,2)(3,4),2);</span>
[ (1,3,2,4), (1,4,2,3) ]
</pre></div>
<p><a id="X804F0F037F06E25E" name="X804F0F037F06E25E"></a></p>
<h4>39.11 <span class="Heading">Normal Structure</span></h4>
<p>For the operations <code class="func">Centralizer</code> (<a href="chap35_mj.html#X7DE33AFC823C7873"><span class="RefLink">35.4-4</span></a>) and <code class="func">Centre</code> (<a href="chap35_mj.html#X847ABE6F781C7FE8"><span class="RefLink">35.4-5</span></a>), see Chapter <a href="chap35_mj.html#X873E502F7D21C39C"><span class="RefLink">35</span></a>.</p>
<p><a id="X87B5370C7DFD401D" name="X87B5370C7DFD401D"></a></p>
<h5>39.11-1 <span class="Heading">Normalizer</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Normalizer</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">‣ Normalizer</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For two groups <var class="Arg">G</var>, <var class="Arg">U</var>, <code class="func">Normalizer</code> computes the normalizer <span class="SimpleMath">\(N_{\textit{G}}(\textit{U})\)</span>, that is, the stabilizer of <var class="Arg">U</var> under the conjugation action of <var class="Arg">G</var>.</p>
<p>For a group <var class="Arg">G</var> and a group element <var class="Arg">g</var>, <code class="func">Normalizer</code> computes <span class="SimpleMath">\(N_{\textit{G}}(\langle \textit{g} \rangle)\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Normalizer(g,Subgroup(g,[(1,2,3)]));</span>
Group([ (1,2,3), (2,3) ])
</pre></div>
<p><a id="X7C4E00297E37AA44" name="X7C4E00297E37AA44"></a></p>
<h5>39.11-2 Core</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Core</code>( <var class="Arg">S</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">S</var> and <var class="Arg">U</var> are groups of elements in the same family, this operation returns the core of <var class="Arg">U</var> in <var class="Arg">S</var>, that is the intersection of all <var class="Arg">S</var>-conjugates of <var class="Arg">U</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Core(g,Subgroup(g,[(1,2,3,4)]));</span>
Group(())
</pre></div>
<p><a id="X7CF497C77B1E8938" name="X7CF497C77B1E8938"></a></p>
<h5>39.11-3 PCore</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PCore</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The <em><var class="Arg">p</var>-core</em> of <var class="Arg">G</var> is the largest normal <var class="Arg">p</var>-subgroup of <var class="Arg">G</var>. It is the core of a Sylow <var class="Arg">p</var>-subgroup of <var class="Arg">G</var>, see <code class="func">Core</code> (<a href="chap39_mj.html#X7C4E00297E37AA44"><span class="RefLink">39.11-2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=QuaternionGroup(12);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PCore(g,2);</span>
Group([ y3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">PCore(g,2) = Core(g,SylowSubgroup(g,2));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">PCore(g,3);</span>
Group([ y*y3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">PCore(g,5);</span>
Group([ ])
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PCore(g,2);</span>
Group([ (1,4)(2,3), (1,2)(3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">PCore(g,2) = Core(g,SylowSubgroup(g,2));</span>
true
</pre></div>
<p><a id="X7BDEA0A98720D1BB" name="X7BDEA0A98720D1BB"></a></p>
<h5>39.11-4 NormalClosure</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalClosure</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The normal closure of <var class="Arg">U</var> in <var class="Arg">G</var> is the smallest normal subgroup of the closure of <var class="Arg">G</var> and <var class="Arg">U</var> which contains <var class="Arg">U</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalClosure(g,Subgroup(g,[(1,2,3)])) = Group([ (1,2,3), (2,3,4) ]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalClosure(g,Group((3,4,5))) = Group([ (3,4,5), (1,5,4), (1,2,5) ]);</span>
true
</pre></div>
<p><a id="X7D25E7DC7834A703" name="X7D25E7DC7834A703"></a></p>
<h5>39.11-5 NormalIntersection</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalIntersection</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes the intersection of <var class="Arg">G</var> and <var class="Arg">U</var>, assuming that <var class="Arg">G</var> is normalized by <var class="Arg">U</var>. This works faster than <code class="code">Intersection</code>, but will not produce the intersection if <var class="Arg">G</var> is not normalized by <var class="Arg">U</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalIntersection(Group((1,2)(3,4),(1,3)(2,4)),Group((1,2,3,4)));</span>
Group([ (1,3)(2,4) ])
</pre></div>
<p><a id="X811B8A4683DDE1F9" name="X811B8A4683DDE1F9"></a></p>
<h5>39.11-6 ComplementClassesRepresentatives</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComplementClassesRepresentatives</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">N</var> be a normal subgroup of <var class="Arg">G</var>. This command returns a set of representatives for the conjugacy classes of complements of <var class="Arg">N</var> in <var class="Arg">G</var>. Complements are subgroups of <var class="Arg">G</var> which intersect trivially with <var class="Arg">N</var> and together with <var class="Arg">N</var> generate <var class="Arg">G</var>.</p>
<p>At the moment only methods for a solvable <var class="Arg">N</var> are available.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ComplementClassesRepresentatives(g,Group((1,2)(3,4),(1,3)(2,4)));</span>
[ Group([ (3,4), (2,4,3) ]) ]
</pre></div>
<p><a id="X8581F4E77B11C610" name="X8581F4E77B11C610"></a></p>
<h5>39.11-7 InfoComplement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoComplement</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>Info class for the complement routines.</p>
<p><a id="X7C39EE3E836D6BC6" name="X7C39EE3E836D6BC6"></a></p>
<h4>39.12 <span class="Heading">Specific and Parametrized Subgroups</span></h4>
<p>The centre of a group (the subgroup of those elements that commute with all other elements of the group) can be computed by the operation <code class="func">Centre</code> (<a href="chap35_mj.html#X847ABE6F781C7FE8"><span class="RefLink">35.4-5</span></a>).</p>
<p><a id="X829759F67D4247CA" name="X829759F67D4247CA"></a></p>
<h5>39.12-1 TrivialSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrivialSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TrivialSubgroup(g);</span>
Group(())
</pre></div>
<p><a id="X7A9A3D5578CE33A0" name="X7A9A3D5578CE33A0"></a></p>
<h5>39.12-2 CommutatorSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CommutatorSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">G</var> and <var class="Arg">H</var> are two groups of elements in the same family, this operation returns the group generated by all commutators <span class="SimpleMath">\([ g, h ] = g^{{-1}} h^{{-1}} g h\)</span> (see <code class="func">Comm</code> (<a href="chap31_mj.html#X80761843831B468E"><span class="RefLink">31.12-3</span></a>)) of elements <span class="SimpleMath">\(g \in \textit{G}\)</span> and <span class="SimpleMath">\(h \in \textit{H}\)</span>, that is the group <span class="SimpleMath">\(\left \langle [ g, h ] \mid g \in \textit{G}, h \in \textit{H} \right \rangle\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CommutatorSubgroup(Group((1,2,3),(1,2)),Group((2,3,4),(3,4)));</span>
Group([ (1,4)(2,3), (1,3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(last);</span>
12
</pre></div>
<p><a id="X7CC17CF179ED7EF2" name="X7CC17CF179ED7EF2"></a></p>
<h5>39.12-3 DerivedSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivedSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The derived subgroup <span class="SimpleMath">\(\textit{G}'\)</span> of <var class="Arg">G</var> is the subgroup generated by all commutators of pairs of elements of <var class="Arg">G</var>. It is normal in <var class="Arg">G</var> and the factor group <span class="SimpleMath">\(\textit{G}/\textit{G}'\)</span> is the largest abelian factor group of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DerivedSubgroup(g) = Group([ (1,3,2), (2,4,3) ]);</span>
true
</pre></div>
<p><a id="X7B10B58F83DDE56E" name="X7B10B58F83DDE56E"></a></p>
<h5>39.12-4 CommutatorLength</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CommutatorLength</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the minimal number <span class="SimpleMath">\(n\)</span> such that each element in the derived subgroup (see <code class="func">DerivedSubgroup</code> (<a href="chap39_mj.html#X7CC17CF179ED7EF2"><span class="RefLink">39.12-3</span></a>)) of the group <var class="Arg">G</var> can be written as a product of (at most) <span class="SimpleMath">\(n\)</span> commutators of elements in <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CommutatorLength( g );</span>
1
</pre></div>
<p><a id="X780552B57C30DD8F" name="X780552B57C30DD8F"></a></p>
<h5>39.12-5 FittingSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FittingSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The Fitting subgroup of a group <var class="Arg">G</var> is its largest nilpotent normal subgroup.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FittingSubgroup(g);</span>
Group([ (1,2)(3,4), (1,4)(2,3) ])
</pre></div>
<p><a id="X788C856C82243274" name="X788C856C82243274"></a></p>
<h5>39.12-6 FrattiniSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FrattiniSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The Frattini subgroup of a group <var class="Arg">G</var> is the intersection of all maximal subgroups of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FrattiniSubgroup(g);</span>
Group(())
</pre></div>
<p><a id="X81D86CCE84193E4F" name="X81D86CCE84193E4F"></a></p>
<h5>39.12-7 PrefrattiniSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrefrattiniSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a Prefrattini subgroup of the finite solvable group <var class="Arg">G</var>.</p>
<p>A factor <span class="SimpleMath">\(M/N\)</span> of <var class="Arg">G</var> is called a Frattini factor if <span class="SimpleMath">\(M/N\)</span> is contained in the Frattini subgroup of <span class="SimpleMath">\(\textit{G}/N\)</span>. A subgroup <span class="SimpleMath">\(P\)</span> is a Prefrattini subgroup of <var class="Arg">G</var> if <span class="SimpleMath">\(P\)</span> covers each Frattini chief factor of <var class="Arg">G</var>, and if for each maximal subgroup of <var class="Arg">G</var> there exists a conjugate maximal subgroup, which contains <span class="SimpleMath">\(P\)</span>. In a finite solvable group <var class="Arg">G</var> the Prefrattini subgroups form a characteristic conjugacy class of subgroups and the intersection of all these subgroups is the Frattini subgroup of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SmallGroup( 60, 7 );</span>
<pc group of size 60 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">P := PrefrattiniSubgroup(G);</span>
Group([ f2 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(P);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNilpotent(P);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Core(G,P);</span>
Group([ ])
<span class="GAPprompt">gap></span> <span class="GAPinput">FrattiniSubgroup(G);</span>
Group([ ])
</pre></div>
<p><a id="X83D5C8B8865C85F1" name="X83D5C8B8865C85F1"></a></p>
<h5>39.12-8 PerfectResiduum</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PerfectResiduum</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the smallest normal subgroup of <var class="Arg">G</var> that has a solvable factor group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PerfectResiduum(Group((1,2,3,4,5),(1,2)));</span>
Group([ (1,3,2), (1,4,3), (3,5,4) ])
</pre></div>
<p><a id="X787F5F14844FAACE" name="X787F5F14844FAACE"></a></p>
<h5>39.12-9 RadicalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RadicalGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the radical of <var class="Arg">G</var>, i.e., the largest solvable normal subgroup of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RadicalGroup(SL(2,5));</span>
<group of 2x2 matrices of size 2 over GF(5)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(last);</span>
2
</pre></div>
<p><a id="X81F647FA83D8854F" name="X81F647FA83D8854F"></a></p>
<h5>39.12-10 Socle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Socle</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The socle of the group <var class="Arg">G</var> is the subgroup generated by all minimal normal subgroups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Socle(g);</span>
Group([ (1,4)(2,3), (1,2)(3,4) ])
</pre></div>
<p><a id="X8440C61080CDAA14" name="X8440C61080CDAA14"></a></p>
<h5>39.12-11 SupersolvableResiduum</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SupersolvableResiduum</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the supersolvable residuum of the group <var class="Arg">G</var>, that is, its smallest normal subgroup <span class="SimpleMath">\(N\)</span> such that the factor group <span class="SimpleMath">\(\textit{G} / N\)</span> is supersolvable.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SupersolvableResiduum(g) = Group([ (1,3)(2,4), (1,4)(2,3) ]);</span>
true
</pre></div>
<p><a id="X796DA805853FAC90" name="X796DA805853FAC90"></a></p>
<h5>39.12-12 PRump</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PRump</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a prime <span class="SimpleMath">\(p\)</span>, the <em><var class="Arg">p</var>-rump</em> of a group <var class="Arg">G</var> is the subgroup <span class="SimpleMath">\(\textit{G}' \textit{G}^{\textit{p}}\)</span>. Unless it equals <var class="Arg">G</var> itself (which is the e.g. the case if <var class="Arg">G</var> is perfect), it is equal to the second term of the <var class="Arg">p</var>-central series of <var class="Arg">G</var>, see <code class="func">PCentralSeries</code> (<a href="chap39_mj.html#X7809B7ED792669F3"><span class="RefLink">39.17-13</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=QuaternionGroup(12);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PRump(g,2) = PCentralSeries(g,2)[2];</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PRump(g,2) = AlternatingGroup(4);</span>
true
</pre></div>
<p><a id="X7FF0BBDD80E8F6BF" name="X7FF0BBDD80E8F6BF"></a></p>
<h4>39.13 <span class="Heading">Sylow Subgroups and Hall Subgroups</span></h4>
<p>With respect to the following <strong class="pkg">GAP</strong> functions, please note that by theorems of P. Hall, a group <span class="SimpleMath">\(G\)</span> is solvable if and only if one of the following conditions holds.</p>
<ol>
<li><p>For each prime <span class="SimpleMath">\(p\)</span> dividing the order of <span class="SimpleMath">\(G\)</span>, there exists a <span class="SimpleMath">\(p\)</span>-complement (see <code class="func">SylowComplement</code> (<a href="chap39_mj.html#X8605F3FE7A3B8E12"><span class="RefLink">39.13-2</span></a>)).</p>
</li>
<li><p>For each set <span class="SimpleMath">\(P\)</span> of primes dividing the order of <span class="SimpleMath">\(G\)</span>, there exists a <span class="SimpleMath">\(P\)</span>-Hall subgroup (see <code class="func">HallSubgroup</code> (<a href="chap39_mj.html#X7EDBA19E828CD584"><span class="RefLink">39.13-3</span></a>)).</p>
</li>
<li><p><span class="SimpleMath">\(G\)</span> has a Sylow system (see <code class="func">SylowSystem</code> (<a href="chap39_mj.html#X832E8E6B8347B13F"><span class="RefLink">39.13-4</span></a>)).</p>
</li>
<li><p><span class="SimpleMath">\(G\)</span> has a complement system (see <code class="func">ComplementSystem</code> (<a href="chap39_mj.html#X87A245E180D27147"><span class="RefLink">39.13-5</span></a>)).</p>
</li>
</ol>
<p><a id="X7AA351308787544C" name="X7AA351308787544C"></a></p>
<h5>39.13-1 SylowSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SylowSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a Sylow <var class="Arg">p</var>-subgroup of the finite group <var class="Arg">G</var>. This is a <var class="Arg">p</var>-subgroup of <var class="Arg">G</var> whose index in <var class="Arg">G</var> is coprime to <var class="Arg">p</var>. <code class="func">SylowSubgroup</code> computes Sylow subgroups via the operation <code class="code">SylowSubgroupOp</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SylowSubgroup(g,2);</span>
Group([ (1,2), (3,4), (1,3)(2,4) ])
</pre></div>
<p><a id="X8605F3FE7A3B8E12" name="X8605F3FE7A3B8E12"></a></p>
<h5>39.13-2 SylowComplement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SylowComplement</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a Sylow <var class="Arg">p</var>-complement of the finite group <var class="Arg">G</var>. This is a subgroup <span class="SimpleMath">\(U\)</span> of order coprime to <var class="Arg">p</var> such that the index <span class="SimpleMath">\([\textit{G}:U]\)</span> is a <var class="Arg">p</var>-power.</p>
<p>At the moment methods exist only if <var class="Arg">G</var> is solvable and <strong class="pkg">GAP</strong> will issue an error if <var class="Arg">G</var> is not solvable.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SylowComplement(g,3);</span>
Group([ (1,2), (3,4), (1,3)(2,4) ])
</pre></div>
<p><a id="X7EDBA19E828CD584" name="X7EDBA19E828CD584"></a></p>
<h5>39.13-3 HallSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HallSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes a <var class="Arg">P</var>-Hall subgroup for a set <var class="Arg">P</var> of primes. This is a subgroup the order of which is only divisible by primes in <var class="Arg">P</var> and whose index is coprime to all primes in <var class="Arg">P</var>. Such a subgroup is unique up to conjugacy if <var class="Arg">G</var> is solvable. The function computes Hall subgroups via the operation <code class="code">HallSubgroupOp</code>.</p>
<p>If <var class="Arg">G</var> is solvable this function always returns a subgroup. If <var class="Arg">G</var> is not solvable this function might return a subgroup (if it is unique up to conjugacy), a list of subgroups (which are representatives of the conjugacy classes in case there are several such classes) or <code class="keyw">fail</code> if no such subgroup exists.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=SmallGroup(60,10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u:=HallSubgroup(h,[2,3]);</span>
Group([ f1, f2, f3 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(u);</span>
12
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=PSL(3,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HallSubgroup(h,[2,3]); </span>
[ <permutation group of size 96 with 6 generators>,
<permutation group of size 96 with 6 generators> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">u := HallSubgroup(h,[3,31]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(u); StructureDescription(u);</span>
93
"C31 : C3"
<span class="GAPprompt">gap></span> <span class="GAPinput">HallSubgroup(h,[5,31]);</span>
fail
</pre></div>
<p><a id="X832E8E6B8347B13F" name="X832E8E6B8347B13F"></a></p>
<h5>39.13-4 SylowSystem</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SylowSystem</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A Sylow system of a group <var class="Arg">G</var> is a set of Sylow subgroups of <var class="Arg">G</var> such that every pair of subgroups from this set commutes as subgroups. Sylow systems exist only for solvable groups. The operation returns <code class="keyw">fail</code> if the group <var class="Arg">G</var> is not solvable.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=SmallGroup(60,10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SylowSystem(h);</span>
[ Group([ f1, f2 ]), Group([ f3 ]), Group([ f4 ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(last,Size);</span>
[ 4, 3, 5 ]
</pre></div>
<p><a id="X87A245E180D27147" name="X87A245E180D27147"></a></p>
<h5>39.13-5 ComplementSystem</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComplementSystem</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A complement system of a group <var class="Arg">G</var> is a set of Hall <span class="SimpleMath">\(p'\)</span>-subgroups of <var class="Arg">G</var>, where <span class="SimpleMath">\(p'\)</span> runs through the subsets of prime factors of <span class="SimpleMath">\(|\textit{G}|\)</span> that omit exactly one prime. Every pair of subgroups from this set commutes as subgroups. Complement systems exist only for solvable groups, therefore <code class="func">ComplementSystem</code> returns <code class="keyw">fail</code> if the group <var class="Arg">G</var> is not solvable.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ComplementSystem(h);</span>
[ Group([ f3, f4 ]), Group([ f1, f2, f4 ]), Group([ f1, f2, f3 ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(last,Size);</span>
[ 15, 20, 12 ]
</pre></div>
<p><a id="X82FE5DFD84F8A3C6" name="X82FE5DFD84F8A3C6"></a></p>
<h5>39.13-6 HallSystem</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HallSystem</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list containing one Hall <span class="SimpleMath">\(P\)</span>-subgroup for each set <span class="SimpleMath">\(P\)</span> of prime divisors of the order of <var class="Arg">G</var>. Hall systems exist only for solvable groups. The operation returns <code class="keyw">fail</code> if the group <var class="Arg">G</var> is not solvable.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">HallSystem(h);</span>
[ Group([ ]), Group([ f1, f2 ]), Group([ f1, f2, f3 ]),
Group([ f1, f2, f3, f4 ]), Group([ f1, f2, f4 ]), Group([ f3 ]),
Group([ f3, f4 ]), Group([ f4 ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(last,Size);</span>
[ 1, 4, 12, 60, 20, 3, 15, 5 ]
</pre></div>
<p><a id="X87AF37E980382499" name="X87AF37E980382499"></a></p>
<h4>39.14 <span class="Heading">Subgroups characterized by prime powers</span></h4>
<p><a id="X7F069ACC83DB3374" name="X7F069ACC83DB3374"></a></p>
<h5>39.14-1 Omega</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Omega</code>( <var class="Arg">G</var>, <var class="Arg">p</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a <var class="Arg">p</var>-group <var class="Arg">G</var>, one defines <span class="SimpleMath">\(\Omega_{\textit{n}}(\textit{G}) = \{ g \in \textit{G} \mid g^{{\textit{p}^{\textit{n}}}} = 1 \}\)</span>. The default value for <var class="Arg">n</var> is <code class="code">1</code>.</p>
<p><em>@At the moment methods exist only for abelian <var class="Arg">G</var> and <var class="Arg">n</var>=1.@</em></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=SmallGroup(16,10);</span>
<pc group of size 16 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Omega(h,2);</span>
Group([ f2, f3, f4 ])
</pre></div>
<p><a id="X83DB33747F069ACC" name="X83DB33747F069ACC"></a></p>
<h5>39.14-2 Agemo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Agemo</code>( <var class="Arg">G</var>, <var class="Arg">p</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a <var class="Arg">p</var>-group <var class="Arg">G</var>, one defines <span class="SimpleMath">\(\mho_{\textit{n}}(G) = \langle g^{{\textit{p}^{\textit{n}}}} \mid g \in \textit{G} \rangle\)</span>. The default value for <var class="Arg">n</var> is <code class="code">1</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Agemo(h,2);Agemo(h,2,2);</span>
Group([ f4 ])
Group([ ])
</pre></div>
<p><a id="X7B75879B8085120A" name="X7B75879B8085120A"></a></p>
<h4>39.15 <span class="Heading">Group Properties</span></h4>
<p>Some properties of groups can be defined not only for groups but also for other structures. For example, nilpotency and solvability make sense also for algebras. Note that these names refer to different definitions for groups and algebras, contrary to the situation with finiteness or commutativity. In such cases, the name of the function for groups got a suffix <code class="code">Group</code> to distinguish different meanings for different structures.</p>
<p>Some functions, such as <code class="func">IsPSolvable</code> (<a href="chap39_mj.html#X81130F9A7CFCF6BF"><span class="RefLink">39.15-24</span></a>) and <code class="func">IsPNilpotent</code> (<a href="chap39_mj.html#X87415A8485FCF510"><span class="RefLink">39.15-25</span></a>), although they are mathematical properties, are not properties in the sense of <strong class="pkg">GAP</strong> (see <a href="chap13_mj.html#X7C701DBF7BAE649A"><span class="RefLink">13.5</span></a> and <a href="chap13_mj.html#X871597447BB998A1"><span class="RefLink">13.7</span></a>), as they depend on a parameter.</p>
<p><a id="X7DA27D338374FD28" name="X7DA27D338374FD28"></a></p>
<h5>39.15-1 IsCyclic</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCyclic</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is <em>cyclic</em> if it can be generated by one element. For a cyclic group, one can compute a generating set consisting of only one element using <code class="func">MinimalGeneratingSet</code> (<a href="chap39_mj.html#X81D15723804771E2"><span class="RefLink">39.22-3</span></a>).</p>
<p><a id="X813C952F80E775D4" name="X813C952F80E775D4"></a></p>
<h5>39.15-2 IsElementaryAbelian</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsElementaryAbelian</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group <var class="Arg">G</var> is elementary abelian if it is commutative and if there is a prime <span class="SimpleMath">\(p\)</span> such that the order of each element in <var class="Arg">G</var> divides <span class="SimpleMath">\(p\)</span>.</p>
<p><a id="X87D062608719F2CD" name="X87D062608719F2CD"></a></p>
<h5>39.15-3 IsNilpotentGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNilpotentGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is <em>nilpotent</em> if the lower central series (see <code class="func">LowerCentralSeriesOfGroup</code> (<a href="chap39_mj.html#X879D55A67DB42676"><span class="RefLink">39.17-11</span></a>) for a definition) reaches the trivial subgroup in a finite number of steps.</p>
<p><a id="X7E3056237C6A5D43" name="X7E3056237C6A5D43"></a></p>
<h5>39.15-4 NilpotencyClassOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NilpotencyClassOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The nilpotency class of a nilpotent group <var class="Arg">G</var> is the number of steps in the lower central series of <var class="Arg">G</var> (see <code class="func">LowerCentralSeriesOfGroup</code> (<a href="chap39_mj.html#X879D55A67DB42676"><span class="RefLink">39.17-11</span></a>));</p>
<p>If <var class="Arg">G</var> is not nilpotent an error is issued.</p>
<p><a id="X8755147280C84DBB" name="X8755147280C84DBB"></a></p>
<h5>39.15-5 IsPerfectGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPerfectGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is <em>perfect</em> if it equals its derived subgroup (see <code class="func">DerivedSubgroup</code> (<a href="chap39_mj.html#X7CC17CF179ED7EF2"><span class="RefLink">39.12-3</span></a>)).</p>
<p><a id="X809C78D5877D31DF" name="X809C78D5877D31DF"></a></p>
<h5>39.15-6 IsSolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSolvableGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is <em>solvable</em> if the derived series (see <code class="func">DerivedSeriesOfGroup</code> (<a href="chap39_mj.html#X7A879948834BD889"><span class="RefLink">39.17-7</span></a>) for a definition) reaches the trivial subgroup in a finite number of steps.</p>
<p>For finite groups this is the same as being polycyclic (see <code class="func">IsPolycyclicGroup</code> (<a href="chap39_mj.html#X7D7456077D3D1B86"><span class="RefLink">39.15-7</span></a>)), and each polycyclic group is solvable, but there are infinite solvable groups that are not polycyclic.</p>
<p><a id="X7D7456077D3D1B86" name="X7D7456077D3D1B86"></a></p>
<h5>39.15-7 IsPolycyclicGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPolycyclicGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is polycyclic if it has a subnormal series with cyclic factors. For finite groups this is the same as if the group is solvable (see <code class="func">IsSolvableGroup</code> (<a href="chap39_mj.html#X809C78D5877D31DF"><span class="RefLink">39.15-6</span></a>)).</p>
<p><a id="X7AADF2E88501B9FF" name="X7AADF2E88501B9FF"></a></p>
<h5>39.15-8 IsSupersolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSupersolvableGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A finite group is <em>supersolvable</em> if it has a normal series with cyclic factors.</p>
<p><a id="X83977EB97A8E2290" name="X83977EB97A8E2290"></a></p>
<h5>39.15-9 IsMonomialGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMonomialGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A finite group is <em>monomial</em> if every irreducible complex character is induced from a linear character of a subgroup.</p>
<p><a id="X7A6685D7819AEC32" name="X7A6685D7819AEC32"></a></p>
<h5>39.15-10 IsSimpleGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSimpleGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is <em>simple</em> if it is nontrivial and has no nontrivial normal subgroups.</p>
<p><a id="X78CC9764803601E7" name="X78CC9764803601E7"></a></p>
<h5>39.15-11 IsAlmostSimpleGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlmostSimpleGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group <var class="Arg">G</var> is <em>almost simple</em> if a nonabelian simple group <span class="SimpleMath">\(S\)</span> exists such that <var class="Arg">G</var> is isomorphic to a subgroup of the automorphism group of <span class="SimpleMath">\(S\)</span> that contains all inner automorphisms of <span class="SimpleMath">\(S\)</span>.</p>
<p>Equivalently, <var class="Arg">G</var> is almost simple if and only if it has a unique minimal normal subgroup <span class="SimpleMath">\(N\)</span> and if <span class="SimpleMath">\(N\)</span> is a nonabelian simple group.</p>
<p>Note that an almost simple group is <em>not</em> defined as an extension of a simple group by outer automorphisms, since we want to exclude extensions of groups of prime order. In particular, a <em>simple</em> group is <em>almost simple</em> if and only if it is nonabelian.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAlmostSimpleGroup( AlternatingGroup( 5 ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAlmostSimpleGroup( SymmetricGroup( 5 ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAlmostSimpleGroup( SymmetricGroup( 3 ) );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAlmostSimpleGroup( SL( 2, 5 ) ); </span>
false
</pre></div>
<p><a id="X7C6AA6897C4409AC" name="X7C6AA6897C4409AC"></a></p>
<h5>39.15-12 <span class="Heading">IsomorphismTypeInfoFiniteSimpleGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismTypeInfoFiniteSimpleGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismTypeInfoFiniteSimpleGroup</code>( <var class="Arg">n</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a finite simple group <var class="Arg">G</var>, <code class="func">IsomorphismTypeInfoFiniteSimpleGroup</code> returns a record with the components <code class="code">name</code>, <code class="code">shortname</code>, <code class="code">series</code>, and possibly <code class="code">parameter</code>, describing the isomorphism type of <var class="Arg">G</var>.</p>
<p>The values of the components <code class="code">name</code>, <code class="code">shortname</code>, and <code class="code">series</code> are strings, <code class="code">name</code> gives name(s) for <var class="Arg">G</var>, <code class="code">shortname</code> gives one name for <var class="Arg">G</var> that is compatible with the naming scheme used in the <strong class="pkg">GAP</strong> packages <strong class="pkg">CTblLib</strong> and <strong class="pkg">AtlasRep</strong> (and in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib_mj.html#biBCCN85">[CCN+85]</a>), and <code class="code">series</code> describes the following series.</p>
<p>(If different characterizations of <var class="Arg">G</var> are possible only one is given by <code class="code">series</code> and <code class="code">parameter</code>, while <code class="code">name</code> may give several names.)</p>
<dl>
<dt><strong class="Mark"><code class="code">"A"</code></strong></dt>
<dd><p>Alternating groups, <code class="code">parameter</code> gives the natural degree.</p>
</dd>
<dt><strong class="Mark"><code class="code">"L"</code></strong></dt>
<dd><p>Linear groups (Chevalley type <span class="SimpleMath">\(A\)</span>), <code class="code">parameter</code> is a list <span class="SimpleMath">\([ n, q ]\)</span> that indicates <span class="SimpleMath">\(L(n,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"2A"</code></strong></dt>
<dd><p>Twisted Chevalley type <span class="SimpleMath">\({}^2A\)</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">\([ n, q ]\)</span> that indicates <span class="SimpleMath">\({}^2A(n,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"B"</code></strong></dt>
<dd><p>Chevalley type <span class="SimpleMath">\(B\)</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">\([n, q ]\)</span> that indicates <span class="SimpleMath">\(B(n,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"2B"</code></strong></dt>
<dd><p>Twisted Chevalley type <span class="SimpleMath">\({}^2B\)</span>, <code class="code">parameter</code> is a value <span class="SimpleMath">\(q\)</span> that indicates <span class="SimpleMath">\({}^2B(2,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"C"</code></strong></dt>
<dd><p>Chevalley type <span class="SimpleMath">\(C\)</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">\([ n, q ]\)</span> that indicates <span class="SimpleMath">\(C(n,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"D"</code></strong></dt>
<dd><p>Chevalley type <span class="SimpleMath">\(D\)</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">\([ n, q ]\)</span> that indicates <span class="SimpleMath">\(D(n,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"2D"</code></strong></dt>
<dd><p>Twisted Chevalley type <span class="SimpleMath">\({}^2D\)</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">\([ n, q ]\)</span> that indicates <span class="SimpleMath">\({}^2D(n,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"3D"</code></strong></dt>
<dd><p>Twisted Chevalley type <span class="SimpleMath">\({}^3D\)</span>, <code class="code">parameter</code> is a value <span class="SimpleMath">\(q\)</span> that indicates <span class="SimpleMath">\({}^3D(4,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"E"</code></strong></dt>
<dd><p>Exceptional Chevalley type <span class="SimpleMath">\(E\)</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">\([ n, q ]\)</span> that indicates <span class="SimpleMath">\(E_n(q)\)</span>. The value of <var class="Arg">n</var> is 6, 7, or 8.</p>
</dd>
<dt><strong class="Mark"><code class="code">"2E"</code></strong></dt>
<dd><p>Twisted exceptional Chevalley type <span class="SimpleMath">\(E_6\)</span>, <code class="code">parameter</code> is a value <span class="SimpleMath">\(q\)</span> that indicates <span class="SimpleMath">\({}^2E_6(q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"F"</code></strong></dt>
<dd><p>Exceptional Chevalley type <span class="SimpleMath">\(F\)</span>, <code class="code">parameter</code> is a value <span class="SimpleMath">\(q\)</span> that indicates <span class="SimpleMath">\(F(4,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"2F"</code></strong></dt>
<dd><p>Twisted exceptional Chevalley type <span class="SimpleMath">\({}^2F\)</span> (Ree groups), <code class="code">parameter</code> is a value <span class="SimpleMath">\(q\)</span> that indicates <span class="SimpleMath">\({}^2F(4,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"G"</code></strong></dt>
<dd><p>Exceptional Chevalley type <span class="SimpleMath">\(G\)</span>, <code class="code">parameter</code> is a value <span class="SimpleMath">\(q\)</span> that indicates <span class="SimpleMath">\(G(2,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"2G"</code></strong></dt>
<dd><p>Twisted exceptional Chevalley type <span class="SimpleMath">\({}^2G\)</span> (Ree groups), <code class="code">parameter</code> is a value <span class="SimpleMath">\(q\)</span> that indicates <span class="SimpleMath">\({}^2G(2,q)\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">"Spor"</code></strong></dt>
<dd><p>Sporadic simple groups, <code class="code">name</code> gives the name.</p>
</dd>
<dt><strong class="Mark"><code class="code">"Z"</code></strong></dt>
<dd><p>Cyclic groups of prime size, <code class="code">parameter</code> gives the size.</p>
</dd>
</dl>
<p>An equal sign in the name denotes different naming schemes for the same group, a tilde sign abstract isomorphisms between groups constructed in a different way.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Group((4,5)(6,7),(1,2,4)(3,5,6)));</span>
rec(
name := "A(1,7) = L(2,7) ~ B(1,7) = O(3,7) ~ C(1,7) = S(2,7) ~ 2A(1,\
7) = U(2,7) ~ A(2,2) = L(3,2)", parameter := [ 2, 7 ], series := "L",
shortname := "L3(2)" )
</pre></div>
<p>For a positive integer <var class="Arg">n</var>, <code class="func">IsomorphismTypeInfoFiniteSimpleGroup</code> returns <code class="keyw">fail</code> if <var class="Arg">n</var> is not the order of a finite simple group, and a record as described for the case of a group <var class="Arg">G</var> otherwise. If more than one simple group of order <var class="Arg">n</var> exists then the result record contains only the <code class="code">name</code> component, a string that lists the two possible isomorphism types of simple groups of this order.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( 5 );</span>
rec( name := "Z(5)", parameter := 5, series := "Z", shortname := "C5"
)
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( 6 );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup(Size(SymplecticGroup(6,3))/2);</span>
rec(
name := "cannot decide from size alone between B(3,3) = O(7,3) and C\
(3,3) = S(6,3)", parameter := [ 3, 3 ] )
</pre></div>
<p><a id="X8492B05B822AC58C" name="X8492B05B822AC58C"></a></p>
<h5>39.15-13 SimpleGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleGroup</code>( <var class="Arg">id</var>[, <var class="Arg">param</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function will construct <strong class="button">an</strong> instance of the specified simple group. Groups are specified via their name in ATLAS style notation, with parameters added if necessary. The intelligence applied to parsing the name is limited, and at the moment no proper extensions can be constructed. For groups who do not have a permutation representation of small degree the ATLASREP package might need to be installed to construct theses groups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SimpleGroup("M(23)");</span>
M23
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(g);</span>
10200960
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SimpleGroup("PSL",3,5);</span>
PSL(3,5)
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(g);</span>
372000
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SimpleGroup("PSp6",2); </span>
PSp(6,2)
</pre></div>
<p><a id="X839CDD8C7AE39FD6" name="X839CDD8C7AE39FD6"></a></p>
<h5>39.15-14 SimpleGroupsIterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimpleGroupsIterator</code>( [<var class="Arg">start</var>[, <var class="Arg">end</var>]] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns an iterator that will run over all simple groups, starting at order <var class="Arg">start</var> if specified, up to order <span class="SimpleMath">\(10^{18}\)</span> (or -- if specified -- order <var class="Arg">end</var>). If the option <var class="Arg">NOPSL2</var> is given, groups of type <span class="SimpleMath">\(PSL_2(q)\)</span> are omitted.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">it:=SimpleGroupsIterator(20000);</span>
<iterator>
<span class="GAPprompt">gap></span> <span class="GAPinput">List([1..8],x->NextIterator(it)); </span>
[ A8, PSL(3,4), PSL(2,37), PSp(4,3), Sz(8), PSL(2,32), PSL(2,41),
PSL(2,43) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">it:=SimpleGroupsIterator(1,2000);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=[];;for i in it do Add(l,i);od;l;</span>
[ A5, PSL(2,7), A6, PSL(2,8), PSL(2,11), PSL(2,13) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">it:=SimpleGroupsIterator(20000,100000:NOPSL2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=[];;for i in it do Add(l,i);od;l;</span>
[ A8, PSL(3,4), PSp(4,3), Sz(8), PSU(3,4), M12 ]
</pre></div>
<p><a id="X872E93F586F54FCE" name="X872E93F586F54FCE"></a></p>
<h5>39.15-15 SmallSimpleGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallSimpleGroup</code>( <var class="Arg">order</var>[, <var class="Arg">i</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">i</var>th simple group of order <var class="Arg">order</var> in the stored list, given in a small-degree permutation representation, or <code class="func">fail</code> (<a href="chap20_mj.html#X8294AAC9860E87E5"><span class="RefLink">20.2-1</span></a>) if no such simple group exists.</p>
<p>If <var class="Arg">i</var> is not given, it defaults to 1. Currently, all simple groups of order less than <span class="SimpleMath">\(10^6\)</span> are available via this function.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallSimpleGroup(60);</span>
A5
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallSimpleGroup(20160,1);</span>
A8
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallSimpleGroup(20160,2);</span>
PSL(3,4)
</pre></div>
<p><a id="X7EB47BF27D8CBF72" name="X7EB47BF27D8CBF72"></a></p>
<h5>39.15-16 AllSmallNonabelianSimpleGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSmallNonabelianSimpleGroups</code>( <var class="Arg">orders</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A list of all nonabelian simple groups whose order lies in the range <var class="Arg">orders</var>.</p>
<p>The groups are given in small-degree permutation representations. The returned list is sorted by ascending group order. Currently, all simple groups of order less than <span class="SimpleMath">\(10^6\)</span> are available via this function.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallNonabelianSimpleGroups([1..1000000]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> StructureDescription);</span>
[ "A5", "PSL(3,2)", "A6", "PSL(2,8)", "PSL(2,11)", "PSL(2,13)",
"PSL(2,17)", "A7", "PSL(2,19)", "PSL(2,16)", "PSL(3,3)",
"PSU(3,3)", "PSL(2,23)", "PSL(2,25)", "M11", "PSL(2,27)",
"PSL(2,29)", "PSL(2,31)", "A8", "PSL(3,4)", "PSL(2,37)", "O(5,3)",
"Sz(8)", "PSL(2,32)", "PSL(2,41)", "PSL(2,43)", "PSL(2,47)",
"PSL(2,49)", "PSU(3,4)", "PSL(2,53)", "M12", "PSL(2,59)",
"PSL(2,61)", "PSU(3,5)", "PSL(2,67)", "J1", "PSL(2,71)", "A9",
"PSL(2,73)", "PSL(2,79)", "PSL(2,64)", "PSL(2,81)", "PSL(2,83)",
"PSL(2,89)", "PSL(3,5)", "M22", "PSL(2,97)", "PSL(2,101)",
"PSL(2,103)", "HJ", "PSL(2,107)", "PSL(2,109)", "PSL(2,113)",
"PSL(2,121)", "PSL(2,125)", "O(5,4)" ]
</pre></div>
<p><a id="X81E22D07871DF37E" name="X81E22D07871DF37E"></a></p>
<h5>39.15-17 IsFinitelyGeneratedGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFinitelyGeneratedGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>tests whether the group <var class="Arg">G</var> can be generated by a finite number of generators. (This property is mainly used to obtain finiteness conditions.)</p>
<p>Note that this is a pure existence statement. Even if a group is known to be generated by a finite number of elements, it can be very hard or even impossible to obtain such a generating set if it is not known.</p>
<p><a id="X8648EDA287829755" name="X8648EDA287829755"></a></p>
<h5>39.15-18 IsSubsetLocallyFiniteGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubsetLocallyFiniteGroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is called locally finite if every finitely generated subgroup is finite. This property checks whether the group <var class="Arg">U</var> is a subset of a locally finite group. This is used to check whether finite generation will imply finiteness, as it does for example for permutation groups.</p>
<p><a id="X8089F18C810B7E3E" name="X8089F18C810B7E3E"></a></p>
<h5>39.15-19 IsPGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A <em><span class="SimpleMath">\(p\)</span>-group</em> is a group in which the order (see <code class="func">Order</code> (<a href="chap31_mj.html#X84F59A2687C62763"><span class="RefLink">31.10-10</span></a>)) of every element is of the form <span class="SimpleMath">\(p^n\)</span> for a prime integer <span class="SimpleMath">\(p\)</span> and a nonnegative integer <span class="SimpleMath">\(n\)</span>. <code class="func">IsPGroup</code> returns <code class="keyw">true</code> if <var class="Arg">G</var> is a <span class="SimpleMath">\(p\)</span>-group, and <code class="keyw">false</code> otherwise.</p>
<p>Finite <span class="SimpleMath">\(p\)</span>-groups are precisely those groups whose order (see <code class="func">Size</code> (<a href="chap30_mj.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>)) is a prime power, and are always nilpotent.</p>
<p>Note that <span class="SimpleMath">\(p\)</span>-groups can also be infinite, and in that case, need not be nilpotent.</p>
<p><a id="X7F232B3F8261CE25" name="X7F232B3F8261CE25"></a></p>
<h5>39.15-20 IsPowerfulPGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPowerfulPGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A finite p-group <var class="Arg">G</var> is said to be a <em>powerful <span class="SimpleMath">\(p\)</span>-group</em> if the commutator subgroup <span class="SimpleMath">\([\textit{G},\textit{G}]\)</span> is contained in <span class="SimpleMath">\(\textit{G}^{p}\)</span> if the prime <span class="SimpleMath">\(p\)</span> is odd, or if <span class="SimpleMath">\([\textit{G},\textit{G}]\)</span> is contained in <span class="SimpleMath">\(\textit{G}^{4}\)</span> if <span class="SimpleMath">\(p = 2\)</span>. The subgroup <span class="SimpleMath">\(\textit{G}^{p}\)</span> is called the first Agemo subgroup, (see <code class="func">Agemo</code> (<a href="chap39_mj.html#X83DB33747F069ACC"><span class="RefLink">39.14-2</span></a>)). <code class="func">IsPowerfulPGroup</code> returns <code class="keyw">true</code> if <var class="Arg">G</var> is a powerful <span class="SimpleMath">\(p\)</span>-group, and <code class="keyw">false</code> otherwise. <em>Note: </em>This function returns <code class="keyw">true</code> if <var class="Arg">G</var> is the trivial group.</p>
<p><a id="X87356BAA7E9E2142" name="X87356BAA7E9E2142"></a></p>
<h5>39.15-21 PrimePGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimePGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">G</var> is a nontrivial <span class="SimpleMath">\(p\)</span>-group (see <code class="func">IsPGroup</code> (<a href="chap39_mj.html#X8089F18C810B7E3E"><span class="RefLink">39.15-19</span></a>)), <code class="func">PrimePGroup</code> returns the prime integer <span class="SimpleMath">\(p\)</span>; if <var class="Arg">G</var> is trivial then <code class="func">PrimePGroup</code> returns <code class="keyw">fail</code>. Otherwise an error is issued.</p>
<p>(One should avoid a common error of writing <code class="code">if IsPGroup(g) then ... PrimePGroup(g) ...</code> where the code represented by dots assumes that <code class="code">PrimePGroup(g)</code> is an integer.)</p>
<p><a id="X863434AD7DDE514B" name="X863434AD7DDE514B"></a></p>
<h5>39.15-22 PClassPGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PClassPGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <span class="SimpleMath">\(p\)</span>-class of a <span class="SimpleMath">\(p\)</span>-group <var class="Arg">G</var> (see <code class="func">IsPGroup</code> (<a href="chap39_mj.html#X8089F18C810B7E3E"><span class="RefLink">39.15-19</span></a>)) is the length of the lower <span class="SimpleMath">\(p\)</span>-central series (see <code class="func">PCentralSeries</code> (<a href="chap39_mj.html#X7809B7ED792669F3"><span class="RefLink">39.17-13</span></a>)) of <var class="Arg">G</var>. If <var class="Arg">G</var> is not a <span class="SimpleMath">\(p\)</span>-group then an error is issued.</p>
<p><a id="X840A4F937ABF15E1" name="X840A4F937ABF15E1"></a></p>
<h5>39.15-23 RankPGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RankPGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a <span class="SimpleMath">\(p\)</span>-group <var class="Arg">G</var> (see <code class="func">IsPGroup</code> (<a href="chap39_mj.html#X8089F18C810B7E3E"><span class="RefLink">39.15-19</span></a>)), <code class="func">RankPGroup</code> returns the <em>rank</em> of <var class="Arg">G</var>, which is defined as the minimal size of a generating system of <var class="Arg">G</var>. If <var class="Arg">G</var> is not a <span class="SimpleMath">\(p\)</span>-group then an error is issued.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=Group((1,2,3,4),(1,3));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PClassPGroup(h);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">RankPGroup(h);</span>
2
</pre></div>
<p><a id="X81130F9A7CFCF6BF" name="X81130F9A7CFCF6BF"></a></p>
<h5>39.15-24 IsPSolvable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPSolvable</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A finite group is <span class="SimpleMath">\(p\)</span>-solvable if every chief factor either has order not divisible by <span class="SimpleMath">\(p\)</span>, or is solvable.</p>
<p><a id="X87415A8485FCF510" name="X87415A8485FCF510"></a></p>
<h5>39.15-25 IsPNilpotent</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPNilpotent</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A group is <span class="SimpleMath">\(p\)</span>-nilpotent if it possesses a normal <span class="SimpleMath">\(p\)</span>-complement.</p>
<p><a id="X7F8264FA796B2B7D" name="X7F8264FA796B2B7D"></a></p>
<h4>39.16 <span class="Heading">Numerical Group Attributes</span></h4>
<p>This section gives only some examples of numerical group attributes, so it should not serve as a collection of all numerical group attributes. The manual contains more such attributes documented in this manual, for example, <code class="func">NrConjugacyClasses</code> (<a href="chap39_mj.html#X8733F87B7E4C9903"><span class="RefLink">39.10-5</span></a>), <code class="func">NilpotencyClassOfGroup</code> (<a href="chap39_mj.html#X7E3056237C6A5D43"><span class="RefLink">39.15-4</span></a>) and others.</p>
<p>Note also that some functions, such as <code class="func">EulerianFunction</code> (<a href="chap39_mj.html#X843E0CCA8351FDF4"><span class="RefLink">39.16-3</span></a>), are mathematical attributes, but not <strong class="pkg">GAP</strong> attributes (see <a href="chap13_mj.html#X7C701DBF7BAE649A"><span class="RefLink">13.5</span></a>) as they are depending on a parameter.</p>
<p><a id="X812827937F403300" name="X812827937F403300"></a></p>
<h5>39.16-1 AbelianInvariants</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianInvariants</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the abelian invariants (also sometimes called primary decomposition) of the commutator factor group of the group <var class="Arg">G</var>. These are given as a list of prime-powers or zeroes and describe the structure of <span class="SimpleMath">\(\textit{G}/\textit{G}'\)</span> as a direct product of cyclic groups of prime power (or infinite) order.</p>
<p>(See <code class="func">IndependentGeneratorsOfAbelianGroup</code> (<a href="chap39_mj.html#X7D1574457B152333"><span class="RefLink">39.22-5</span></a>) to obtain actual generators).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2),(5,6));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(g);</span>
[ 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=FreeGroup(2);;h:=h/[h.1^3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(h);</span>
[ 0, 3 ]
</pre></div>
<p><a id="X7D44470C7DA59C1C" name="X7D44470C7DA59C1C"></a></p>
<h5>39.16-2 Exponent</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Exponent</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The exponent <span class="SimpleMath">\(e\)</span> of a group <var class="Arg">G</var> is the lcm of the orders of its elements, that is, <span class="SimpleMath">\(e\)</span> is the smallest integer such that <span class="SimpleMath">\(g^e = 1\)</span> for all <span class="SimpleMath">\(g \in \textit{G}\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Exponent(g);</span>
12
</pre></div>
<p><a id="X843E0CCA8351FDF4" name="X843E0CCA8351FDF4"></a></p>
<h5>39.16-3 EulerianFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EulerianFunction</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the number of <var class="Arg">n</var>-tuples <span class="SimpleMath">\((g_1, g_2, \ldots, g_n)\)</span> of elements of the group <var class="Arg">G</var> that generate the whole group <var class="Arg">G</var>. The elements of such an <var class="Arg">n</var>-tuple need not be different.</p>
<p>In <a href="chapBib_mj.html#biBHal36">[Hal36]</a>, the notation <span class="SimpleMath">\(\phi_{\textit{n}}(\textit{G})\)</span> is used for the value returned by <code class="func">EulerianFunction</code>, and the quotient of <span class="SimpleMath">\(\phi_{\textit{n}}(\textit{G})\)</span> by the order of the automorphism group of <var class="Arg">G</var> is called <span class="SimpleMath">\(d_{\textit{n}}(\textit{G})\)</span>. If <var class="Arg">G</var> is a nonabelian simple group then <span class="SimpleMath">\(d_{\textit{n}}(\textit{G})\)</span> is the greatest number <span class="SimpleMath">\(d\)</span> for which the direct product of <span class="SimpleMath">\(d\)</span> groups isomorphic with <var class="Arg">G</var> can be generated by <var class="Arg">n</var> elements.</p>
<p>If the Library of Tables of Marks (see Chapter <a href="chap70_mj.html#X84DBFB8287C3F1B4"><span class="RefLink">70</span></a>) covers the group <var class="Arg">G</var>, you may also use <code class="func">EulerianFunctionByTom</code> (<a href="chap70_mj.html#X7B1C1A7C867A4082"><span class="RefLink">70.9-9</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">EulerianFunction( g, 2 );</span>
432
</pre></div>
<p><a id="X7AEDEDF67CFED672" name="X7AEDEDF67CFED672"></a></p>
<h4>39.17 <span class="Heading">Subgroup Series</span></h4>
<p>In group theory many subgroup series are considered, and <strong class="pkg">GAP</strong> provides commands to compute them. In the following sections, there is always a series <span class="SimpleMath">\(G = U_1 > U_2 > \cdots > U_m = \langle 1 \rangle\)</span> of subgroups considered. A series also may stop without reaching <span class="SimpleMath">\(G\)</span> or <span class="SimpleMath">\(\langle 1 \rangle\)</span>.</p>
<p>A series is called <em>subnormal</em> if every <span class="SimpleMath">\(U_{{i+1}}\)</span> is normal in <span class="SimpleMath">\(U_i\)</span>.</p>
<p>A series is called <em>normal</em> if every <span class="SimpleMath">\(U_i\)</span> is normal in <span class="SimpleMath">\(G\)</span>.</p>
<p>A series of normal subgroups is called <em>central</em> if <span class="SimpleMath">\(U_i/U_{{i+1}}\)</span> is central in <span class="SimpleMath">\(G / U_{{i+1}}\)</span>.</p>
<p>We call a series <em>refinable</em> if intermediate subgroups can be added to the series without destroying the properties of the series.</p>
<p>Unless explicitly declared otherwise, all subgroup series are descending. That is they are stored in decreasing order.</p>
<p><a id="X7BDD116F7833800F" name="X7BDD116F7833800F"></a></p>
<h5>39.17-1 ChiefSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChiefSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is a series of normal subgroups of <var class="Arg">G</var> which cannot be refined further. That is there is no normal subgroup <span class="SimpleMath">\(N\)</span> of <var class="Arg">G</var> with <span class="SimpleMath">\(U_i > N > U_{{i+1}}\)</span>. This attribute returns <em>one</em> chief series (of potentially many possibilities).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ChiefSeries(g);</span>
[ Group([ (1,2,3,4), (1,2) ]),
Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]),
Group([ (1,4)(2,3), (1,3)(2,4) ]), Group(()) ]
</pre></div>
<p><a id="X7AC93E977AC9ED58" name="X7AC93E977AC9ED58"></a></p>
<h5>39.17-2 ChiefSeriesThrough</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChiefSeriesThrough</code>( <var class="Arg">G</var>, <var class="Arg">l</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is a chief series of the group <var class="Arg">G</var> going through the normal subgroups in the list <var class="Arg">l</var>, which must be a list of normal subgroups of <var class="Arg">G</var> contained in each other, sorted by descending size. This attribute returns <em>one</em> chief series (of potentially many possibilities).</p>
<p><a id="X8724E15F81B51173" name="X8724E15F81B51173"></a></p>
<h5>39.17-3 ChiefSeriesUnderAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChiefSeriesUnderAction</code>( <var class="Arg">H</var>, <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a series of normal subgroups of <var class="Arg">G</var> which are invariant under <var class="Arg">H</var> such that the series cannot be refined any further. <var class="Arg">G</var> must be a subgroup of <var class="Arg">H</var>. This attribute returns <em>one</em> such series (of potentially many possibilities).</p>
<p><a id="X7A0E7A8B8495B79D" name="X7A0E7A8B8495B79D"></a></p>
<h5>39.17-4 SubnormalSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubnormalSeries</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">U</var> is a subgroup of <var class="Arg">G</var> this operation returns a subnormal series that descends from <var class="Arg">G</var> to a subnormal subgroup <span class="SimpleMath">\(V \geq \)</span><var class="Arg">U</var>. If <var class="Arg">U</var> is subnormal, <span class="SimpleMath">\(V =\)</span> <var class="Arg">U</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=SubnormalSeries(g,Group((1,2)(3,4))) =</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ Group([ (1,2,3,4), (1,2) ]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Group([ (1,2)(3,4), (1,3)(2,4) ]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Group([ (1,2)(3,4) ]) ];</span>
true
</pre></div>
<p><a id="X81CDCBD67BC98A5A" name="X81CDCBD67BC98A5A"></a></p>
<h5>39.17-5 CompositionSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompositionSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A composition series is a subnormal series which cannot be refined. This attribute returns <em>one</em> composition series (of potentially many possibilities).</p>
<p><a id="X82C0D0217ACB2042" name="X82C0D0217ACB2042"></a></p>
<h5>39.17-6 DisplayCompositionSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayCompositionSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Displays a composition series of <var class="Arg">G</var> in a nice way, identifying the simple factors.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CompositionSeries(g);</span>
[ Group([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]),
Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]),
Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,3)(2,4) ]), Group(())
]
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayCompositionSeries(Group((1,2,3,4,5,6,7),(1,2)));</span>
G (2 gens, size 5040)
| Z(2)
S (5 gens, size 2520)
| A(7)
1 (0 gens, size 1)
</pre></div>
<p><a id="X7A879948834BD889" name="X7A879948834BD889"></a></p>
<h5>39.17-7 DerivedSeriesOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivedSeriesOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The derived series of a group is obtained by <span class="SimpleMath">\(U_{{i+1}} = U_i'\)</span>. It stops if <span class="SimpleMath">\(U_i\)</span> is perfect.</p>
<p><a id="X7A9AA1577CEC891F" name="X7A9AA1577CEC891F"></a></p>
<h5>39.17-8 DerivedLength</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivedLength</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The derived length of a group is the number of steps in the derived series. (As there is always the group, it is the series length minus 1.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(DerivedSeriesOfGroup(g),Size);</span>
[ 24, 12, 4, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DerivedLength(g);</span>
3
</pre></div>
<p><a id="X83F057E5791944D6" name="X83F057E5791944D6"></a></p>
<h5>39.17-9 <span class="Heading">ElementaryAbelianSeries</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementaryAbelianSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementaryAbelianSeriesLargeSteps</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementaryAbelianSeries</code>( <var class="Arg">list</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a series of normal subgroups of <span class="SimpleMath">\(G\)</span> such that all factors are elementary abelian. If the group is not solvable (and thus no such series exists) it returns <code class="keyw">fail</code>.</p>
<p>The variant <code class="func">ElementaryAbelianSeriesLargeSteps</code> tries to make the steps in this series large (by eliminating intermediate subgroups if possible) at a small additional cost.</p>
<p>In the third variant, an elementary abelian series through the given series of normal subgroups in the list <var class="Arg">list</var> is constructed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(ElementaryAbelianSeries(g),Size);</span>
[ 24, 12, 4, 1 ]
</pre></div>
<p><a id="X782BD7A47D6B6503" name="X782BD7A47D6B6503"></a></p>
<h5>39.17-10 InvariantElementaryAbelianSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InvariantElementaryAbelianSeries</code>( <var class="Arg">G</var>, <var class="Arg">morph</var>[, <var class="Arg">N</var>[, <var class="Arg">fine</var>]] )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a (solvable) group <var class="Arg">G</var> and a list of automorphisms <var class="Arg">morph</var> of <var class="Arg">G</var>, this command finds a normal series of <var class="Arg">G</var> with elementary abelian factors such that every group in this series is invariant under every automorphism in <var class="Arg">morph</var>.</p>
<p>If a normal subgroup <var class="Arg">N</var> of <var class="Arg">G</var> which is invariant under <var class="Arg">morph</var> is given, this series is chosen to contain <var class="Arg">N</var>. No tests are performed to check the validity of the arguments.</p>
<p>The series obtained will be constructed to prefer large steps unless <var class="Arg">fine</var> is given as <code class="keyw">true</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,3));</span>
Group([ (1,2,3,4), (1,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=GroupHomomorphismByImages(g,g,GeneratorsOfGroup(g),</span>
<span class="GAPprompt">></span> <span class="GAPinput">[(1,4,3,2),(1,4)(2,3)]);</span>
[ (1,2,3,4), (1,3) ] -> [ (1,4,3,2), (1,4)(2,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">InvariantElementaryAbelianSeries(g,[hom]);</span>
[ Group([ (1,2,3,4), (1,3) ]), Group([ (1,3)(2,4) ]), Group(()) ]
</pre></div>
<p><a id="X879D55A67DB42676" name="X879D55A67DB42676"></a></p>
<h5>39.17-11 LowerCentralSeriesOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowerCentralSeriesOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The lower central series of a group <var class="Arg">G</var> is defined as <span class="SimpleMath">\(U_{{i+1}}:= [\textit{G}, U_i]\)</span>. It is a central series of normal subgroups. The name derives from the fact that <span class="SimpleMath">\(U_i\)</span> is contained in the <span class="SimpleMath">\(i\)</span>-th step subgroup of any central series.</p>
<p><a id="X8428592E8773CD7B" name="X8428592E8773CD7B"></a></p>
<h5>39.17-12 UpperCentralSeriesOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UpperCentralSeriesOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The upper central series of a group <var class="Arg">G</var> is defined as an ending series <span class="SimpleMath">\(U_i / U_{{i+1}}:= Z(\textit{G}/U_{{i+1}})\)</span>. It is a central series of normal subgroups. The name derives from the fact that <span class="SimpleMath">\(U_i\)</span> contains every <span class="SimpleMath">\(i\)</span>-th step subgroup of a central series.</p>
<p><a id="X7809B7ED792669F3" name="X7809B7ED792669F3"></a></p>
<h5>39.17-13 PCentralSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PCentralSeries</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The <var class="Arg">p</var>-central series of <var class="Arg">G</var> is defined by <span class="SimpleMath">\(U_1:= \textit{G}\)</span>, <span class="SimpleMath">\(U_i:= [\textit{G}, U_{{i-1}}] U_{{i-1}}^{\textit{p}}\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=QuaternionGroup(12);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PCentralSeries(g,2);</span>
[ <pc group of size 12 with 3 generators>, Group([ y3, y*y3 ]), Group([ y*y3 ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PCentralSeries(g,2);</span>
[ Sym( [ 1 .. 4 ] ), Group([ (1,2,3), (2,3,4) ]) ]
</pre></div>
<p><a id="X82A34BD681F24A94" name="X82A34BD681F24A94"></a></p>
<h5>39.17-14 JenningsSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ JenningsSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a <span class="SimpleMath">\(p\)</span>-group <var class="Arg">G</var>, this function returns its Jennings series. This series is defined by setting <span class="SimpleMath">\(G_1 = \textit{G}\)</span> and for <span class="SimpleMath">\(i \geq 0\)</span>, <span class="SimpleMath">\(G_{{i+1}} = [G_i,\textit{G}] G_j^p\)</span>, where <span class="SimpleMath">\(j\)</span> is the smallest integer <span class="SimpleMath">\(> i/p\)</span>.</p>
<p><a id="X7C08A8B77EC09CFF" name="X7C08A8B77EC09CFF"></a></p>
<h5>39.17-15 DimensionsLoewyFactors</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DimensionsLoewyFactors</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This operation computes the dimensions of the factors of the Loewy series of <var class="Arg">G</var>. (See <a href="chapBib_mj.html#biBHup82">[HB82, p. 157]</a> for the slightly complicated definition of the Loewy Series.)</p>
<p>The dimensions are computed via the <code class="func">JenningsSeries</code> (<a href="chap39_mj.html#X82A34BD681F24A94"><span class="RefLink">39.17-14</span></a>) without computing the Loewy series itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:= SmallGroup( 3^6, 100 );</span>
<pc group of size 729 with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">JenningsSeries( G );</span>
[ <pc group of size 729 with 6 generators>, Group([ f3, f4, f5, f6 ]),
Group([ f4, f5, f6 ]), Group([ f5, f6 ]), Group([ f5, f6 ]),
Group([ f5, f6 ]), Group([ f6 ]), Group([ f6 ]), Group([ f6 ]),
Group([ <identity> of ... ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DimensionsLoewyFactors(G);</span>
[ 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26,
27, 27, 27, 27, 27, 27, 27, 27, 27, 26, 25, 23, 22, 20, 19, 17, 16,
14, 13, 11, 10, 8, 7, 5, 4, 2, 1 ]
</pre></div>
<p><a id="X84112774812180DD" name="X84112774812180DD"></a></p>
<h5>39.17-16 AscendingChain</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AscendingChain</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function computes an ascending chain of subgroups from <var class="Arg">U</var> to <var class="Arg">G</var>. This chain is given as a list whose first entry is <var class="Arg">U</var> and the last entry is <var class="Arg">G</var>. The function tries to make the links in this chain small.</p>
<p>The option <code class="code">refineIndex</code> can be used to give a bound for refinements of steps to avoid <strong class="pkg">GAP</strong> trying to enforce too small steps. The option <code class="code">cheap</code> (if set to <code class="keyw">true</code>) will overall limit the amount of heuristic searches.</p>
<p><a id="X7C5029EE86D7FC96" name="X7C5029EE86D7FC96"></a></p>
<h5>39.17-17 IntermediateGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntermediateGroup</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This routine tries to find a subgroup <span class="SimpleMath">\(E\)</span> of <var class="Arg">G</var>, such that <span class="SimpleMath">\(\textit{G} > E > \textit{U}\)</span> holds. If <var class="Arg">U</var> is maximal in <var class="Arg">G</var>, the function returns <code class="keyw">fail</code>. This is done by finding minimal blocks for the operation of <var class="Arg">G</var> on the right cosets of <var class="Arg">U</var>.</p>
<p><a id="X781661FB78DC83B5" name="X781661FB78DC83B5"></a></p>
<h5>39.17-18 IntermediateSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntermediateSubgroups</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 subgroups of <var class="Arg">G</var> that properly contain <var class="Arg">U</var>; that is all subgroups between <var class="Arg">G</var> and <var class="Arg">U</var>. It returns a record with a component <code class="code">subgroups</code>, which is a list of these subgroups, as well as a component <code class="code">inclusions</code>, which lists all maximality inclusions among these subgroups. A maximality inclusion is given as a list <span class="SimpleMath">\([i, j]\)</span> indicating that the subgroup number <span class="SimpleMath">\(i\)</span> is a maximal subgroup of the subgroup number <span class="SimpleMath">\(j\)</span>, the numbers <span class="SimpleMath">\(0\)</span> and <span class="SimpleMath">\(1 +\)</span> <code class="code">Length(subgroups)</code> are used to denote <var class="Arg">U</var> and <var class="Arg">G</var>, respectively.</p>
<p><a id="X84091B0A7E401E2B" name="X84091B0A7E401E2B"></a></p>
<h4>39.18 <span class="Heading">Factor Groups</span></h4>
<p><a id="X80FC390C7F38A13F" name="X80FC390C7F38A13F"></a></p>
<h5>39.18-1 NaturalHomomorphismByNormalSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalHomomorphismByNormalSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">N</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">‣ NaturalHomomorphismByNormalSubgroupNC</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a homomorphism from <var class="Arg">G</var> to another group whose kernel is <var class="Arg">N</var>. <strong class="pkg">GAP</strong> will try to select the image group as to make computations in it as efficient as possible. As the factor group <span class="SimpleMath">\(\textit{G}/\textit{N}\)</span> can be identified with the image of <var class="Arg">G</var> this permits efficient computations in the factor group. The homomorphism returned is not necessarily surjective, so <code class="func">ImagesSource</code> (<a href="chap32_mj.html#X7D23C1CE863DACD8"><span class="RefLink">32.4-1</span></a>) should be used instead of <code class="func">Range</code> (<a href="chap32_mj.html#X7B6FD7277CDE9FCB"><span class="RefLink">32.3-7</span></a>) to get a group isomorphic to the factor group. The <code class="code">NC</code> variant does not check whether <var class="Arg">N</var> is normal in <var class="Arg">G</var>.</p>
<p><a id="X7E6EED0185B27C48" name="X7E6EED0185B27C48"></a></p>
<h5>39.18-2 FactorGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FactorGroup</code>( <var class="Arg">G</var>, <var class="Arg">N</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">‣ FactorGroupNC</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the image of the <code class="code">NaturalHomomorphismByNormalSubgroup(<var class="Arg">G</var>,<var class="Arg">N</var>)</code>. The homomorphism will be returned by calling the function <code class="code">NaturalHomomorphism</code> on the result. The <code class="code">NC</code> version does not test whether <var class="Arg">N</var> is normal in <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;n:=Subgroup(g,[(1,2)(3,4),(1,3)(2,4)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=NaturalHomomorphismByNormalSubgroup(g,n);</span>
[ (1,2,3,4), (1,2) ] -> [ f1*f2, f1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(ImagesSource(hom));</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorGroup(g,n);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(last);</span>
"S3"
</pre></div>
<p><a id="X7816FA867BF1B8ED" name="X7816FA867BF1B8ED"></a></p>
<h5>39.18-3 CommutatorFactorGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CommutatorFactorGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the commutator factor group <span class="SimpleMath">\(\textit{G}/\textit{G}'\)</span> of the group <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CommutatorFactorGroup(g);</span>
Group([ f1 ])
</pre></div>
<p><a id="X7BB93B9778C5A0B2" name="X7BB93B9778C5A0B2"></a></p>
<h5>39.18-4 MaximalAbelianQuotient</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalAbelianQuotient</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an epimorphism from <var class="Arg">G</var> onto the maximal abelian quotient of <var class="Arg">G</var>. The kernel of this epimorphism is the derived subgroup of <var class="Arg">G</var>, see <code class="func">DerivedSubgroup</code> (<a href="chap39_mj.html#X7CC17CF179ED7EF2"><span class="RefLink">39.12-3</span></a>).</p>
<p><a id="X7FC83E4C783572E7" name="X7FC83E4C783572E7"></a></p>
<h5>39.18-5 HasAbelianFactorGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HasAbelianFactorGroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests whether <var class="Arg">G</var> <span class="SimpleMath">\(/\)</span> <var class="Arg">N</var> is abelian (without explicitly constructing the factor group and without testing whether <var class="Arg">N</var> is in fact a normal subgroup).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">HasAbelianFactorGroup(g,n);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">HasAbelianFactorGroup(DerivedSubgroup(g),n);</span>
true
</pre></div>
<p><a id="X7FAC018680B766B7" name="X7FAC018680B766B7"></a></p>
<h5>39.18-6 HasElementaryAbelianFactorGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HasElementaryAbelianFactorGroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests whether <var class="Arg">G</var> <span class="SimpleMath">\(/\)</span> <var class="Arg">N</var> is elementary abelian (without explicitly constructing the factor group and without testing whether <var class="Arg">N</var> is in fact a normal subgroup).</p>
<p><a id="X822A3AB27919BC1E" name="X822A3AB27919BC1E"></a></p>
<h5>39.18-7 CentralizerModulo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CentralizerModulo</code>( <var class="Arg">G</var>, <var class="Arg">N</var>, <var class="Arg">elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Computes the full preimage of the centralizer <span class="SimpleMath">\(C_{{\textit{G}/\textit{N}}}(\textit{elm} \cdot \textit{N})\)</span> in <var class="Arg">G</var> (without necessarily constructing the factor group).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CentralizerModulo(g,n,(1,2));</span>
Group([ (3,4), (1,3)(2,4), (1,4)(2,3) ])
</pre></div>
<p><a id="X7D8EFB2F85AA24EE" name="X7D8EFB2F85AA24EE"></a></p>
<h4>39.19 <span class="Heading">Sets of Subgroups</span></h4>
<p><a id="X7DDE67C67E871336" name="X7DDE67C67E871336"></a></p>
<h5>39.19-1 ConjugacyClassSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClassSubgroups</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>generates the conjugacy class of subgroups of <var class="Arg">G</var> with representative <var class="Arg">U</var>. This class is an external set, so functions such as <code class="func">Representative</code> (<a href="chap30_mj.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>), (which returns <var class="Arg">U</var>), <code class="func">ActingDomain</code> (<a href="chap41_mj.html#X7B9DB15D80CE28B4"><span class="RefLink">41.12-3</span></a>) (which returns <var class="Arg">G</var>), <code class="func">StabilizerOfExternalSet</code> (<a href="chap41_mj.html#X7BAFF02B7D6DF9F2"><span class="RefLink">41.12-10</span></a>) (which returns the normalizer of <var class="Arg">U</var>), and <code class="func">AsList</code> (<a href="chap30_mj.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) work for it.</p>
<p>(The use of the <code class="code">[]</code> list access to select elements of the class is considered obsolescent and will be removed in future versions. Use <code class="func">ClassElementLattice</code> (<a href="chap39_mj.html#X78928A3582882BFD"><span class="RefLink">39.20-2</span></a>) instead.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;IsNaturalSymmetricGroup(g);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cl:=ConjugacyClassSubgroups(g,Subgroup(g,[(1,2)]));</span>
Group( [ (1,2) ] )^G
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(cl);</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassElementLattice(cl,4);</span>
Group([ (2,3) ])
</pre></div>
<p><a id="X7C5BBF487977B8CD" name="X7C5BBF487977B8CD"></a></p>
<h5>39.19-2 IsConjugacyClassSubgroupsRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsConjugacyClassSubgroupsRep</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsConjugacyClassSubgroupsByStabilizerRep</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Is the representation <strong class="pkg">GAP</strong> uses for conjugacy classes of subgroups. It can be used to check whether an object is a class of subgroups. The second representation <code class="func">IsConjugacyClassSubgroupsByStabilizerRep</code> in addition is an external orbit by stabilizer and will compute its elements via a transversal of the stabilizer.</p>
<p><a id="X7E986BF48393113A" name="X7E986BF48393113A"></a></p>
<h5>39.19-3 ConjugacyClassesSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClassesSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute returns a list of all conjugacy classes of subgroups of the group <var class="Arg">G</var>. It also is applicable for lattices of subgroups (see <code class="func">LatticeSubgroups</code> (<a href="chap39_mj.html#X7B104E2C86166188"><span class="RefLink">39.20-1</span></a>)). The order in which the classes are listed depends on the method chosen by <strong class="pkg">GAP</strong>. For each class of subgroups, a representative can be accessed using <code class="func">Representative</code> (<a href="chap30_mj.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ConjugacyClassesSubgroups(g);</span>
[ Group( () )^G, Group( [ (1,3)(2,4) ] )^G, Group( [ (3,4) ] )^G,
Group( [ (2,4,3) ] )^G, Group( [ (1,4)(2,3), (1,3)(2,4) ] )^G,
Group( [ (3,4), (1,2)(3,4) ] )^G,
Group( [ (1,3,2,4), (1,2)(3,4) ] )^G, Group( [ (3,4), (2,4,3) ] )^G,
Group( [ (1,4)(2,3), (1,3)(2,4), (3,4) ] )^G,
Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3) ] )^G,
Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ] )^G ]
</pre></div>
<p><a id="X8486C25380853F9B" name="X8486C25380853F9B"></a></p>
<h5>39.19-4 ConjugacyClassesMaximalSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClassesMaximalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the conjugacy classes of maximal subgroups of <var class="Arg">G</var>. Representatives of the classes can be computed directly by <code class="func">MaximalSubgroupClassReps</code> (<a href="chap39_mj.html#X798BF55C837DB188"><span class="RefLink">39.19-5</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ConjugacyClassesMaximalSubgroups(g);</span>
[ Group( [ (2,4,3), (1,4)(2,3), (1,3)(2,4) ] )^G,
Group( [ (3,4), (1,4)(2,3), (1,3)(2,4) ] )^G,
Group( [ (3,4), (2,4,3) ] )^G ]
</pre></div>
<p><a id="X798BF55C837DB188" name="X798BF55C837DB188"></a></p>
<h5>39.19-5 MaximalSubgroupClassReps</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalSubgroupClassReps</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of conjugacy representatives of the maximal subgroups of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalSubgroupClassReps(g);</span>
[ Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]), Group([ (3,4), (1,4)
(2,3), (1,3)(2,4) ]), Group([ (3,4), (2,4,3) ]) ]
</pre></div>
<p><a id="X85DAFB7582A88463" name="X85DAFB7582A88463"></a></p>
<h5>39.19-6 LowIndexSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowIndexSubgroups</code>( <var class="Arg">G</var>, <var class="Arg">index</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The operation <code class="func">LowIndexSubgroups</code> computes representatives of the conjugacy classes of subgroups of the group <var class="Arg">G</var> that index less than or equal to <var class="Arg">index</var>.</p>
<p>For finitely presented groups this operation simply defaults to <code class="func">LowIndexSubgroupsFpGroup</code> (<a href="chap47_mj.html#X85C5151380E19122"><span class="RefLink">47.10-1</span></a>). In other cases, it uses repeated calculation of maximal subgroups.</p>
<p>The function <code class="func">LowLayerSubgroups</code> (<a href="chap39_mj.html#X87BE970D7B18E2C5"><span class="RefLink">39.20-6</span></a>) works similar but does not bound the index, but instead considers up to <var class="Arg">layer</var>-th maximal subgroups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=TransitiveGroup(18,950);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=LowIndexSubgroups(g,20);;Collected(List(l,x->Index(g,x)));</span>
[ [ 1, 1 ], [ 2, 1 ], [ 5, 1 ], [ 6, 1 ], [ 10, 2 ], [ 12, 3 ], [ 15, 1 ],
[ 16, 2 ], [ 18, 1 ], [ 20, 9 ] ]
</pre></div>
<p><a id="X80399CD4870FFC4B" name="X80399CD4870FFC4B"></a></p>
<h5>39.19-7 AllSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a finite group <var class="Arg">G</var> <code class="func">AllSubgroups</code> returns a list of all subgroups of <var class="Arg">G</var>, intended primarily for use in class for small examples. This list will quickly get very long and in general use of <code class="func">ConjugacyClassesSubgroups</code> (<a href="chap39_mj.html#X7E986BF48393113A"><span class="RefLink">39.19-3</span></a>) is recommended.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllSubgroups(SymmetricGroup(3));</span>
[ Group(()), Group([ (2,3) ]), Group([ (1,2) ]), Group([ (1,3) ]),
Group([ (1,2,3) ]), Group([ (1,2,3), (2,3) ]) ]
</pre></div>
<p><a id="X861CD8DA790D81C2" name="X861CD8DA790D81C2"></a></p>
<h5>39.19-8 MaximalSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of all maximal subgroups of <var class="Arg">G</var>. This may take up much space, therefore the command should be avoided if possible. See <code class="func">ConjugacyClassesMaximalSubgroups</code> (<a href="chap39_mj.html#X8486C25380853F9B"><span class="RefLink">39.19-4</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalSubgroups(Group((1,2,3),(1,2)));</span>
[ Group([ (1,2,3) ]), Group([ (2,3) ]), Group([ (1,2) ]),
Group([ (1,3) ]) ]
</pre></div>
<p><a id="X80237A847E24E6CF" name="X80237A847E24E6CF"></a></p>
<h5>39.19-9 NormalSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of all normal subgroups of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);;NormalSubgroups(g);</span>
[ Sym( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,2)
(3,4) ]), Group(()) ]
</pre></div>
<p>The algorithm for the computation of normal subgroups is described in <a href="chapBib_mj.html#biBHulpke98">[Hul98]</a>.</p>
<p><a id="X82ECAA427C987318" name="X82ECAA427C987318"></a></p>
<h5>39.19-10 MaximalNormalSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalNormalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is a list containing those proper normal subgroups of the group <var class="Arg">G</var> that are maximal among the proper normal subgroups. Gives error if <var class="Arg">G</var>/<var class="Arg">G'</var> is infinite, yielding infinitely many maximal normal subgroups. Note, that the maximal normal subgroups of a group <var class="Arg">G</var> can be computed more efficiently if the character table of <var class="Arg">G</var> is known or if <var class="Arg">G</var> is known to be abelian or solvable (even if infinite). So if the character table is needed, anyhow, or <var class="Arg">G</var> is suspected to be abelian or solvable, then these should be computed before computing the maximal normal subgroups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalNormalSubgroups( g );</span>
[ Group([ (1,2,3), (2,3,4) ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeGroup("x", "y");; x := f.1;; y := f.2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(MaximalNormalSubgroups(f/[x^2, y^2]), GeneratorsOfGroup);</span>
[ [ x, y*x*y^-1 ], [ y, x*y*x^-1 ], [ y*x^-1 ] ]
</pre></div>
<p><a id="X86FDD9BA819F5644" name="X86FDD9BA819F5644"></a></p>
<h5>39.19-11 MinimalNormalSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalNormalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is a list containing those nontrivial normal subgroups of the group <var class="Arg">G</var> that are minimal among the nontrivial normal subgroups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalNormalSubgroups( g );</span>
[ Group([ (1,4)(2,3), (1,3)(2,4) ]) ]
</pre></div>
<p><a id="X7FA267497CFC0550" name="X7FA267497CFC0550"></a></p>
<h4>39.20 <span class="Heading">Subgroup Lattice</span></h4>
<p><a id="X7B104E2C86166188" name="X7B104E2C86166188"></a></p>
<h5>39.20-1 LatticeSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LatticeSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the lattice of subgroups of the group <var class="Arg">G</var>. This lattice has the conjugacy classes of subgroups as attribute <code class="func">ConjugacyClassesSubgroups</code> (<a href="chap39_mj.html#X7E986BF48393113A"><span class="RefLink">39.19-3</span></a>) and permits one to test maximality/minimality relations.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=LatticeSubgroups(g);</span>
<subgroup lattice of Sym( [ 1 .. 4 ] ), 11 classes, 30 subgroups>
<span class="GAPprompt">gap></span> <span class="GAPinput">ConjugacyClassesSubgroups(l);</span>
[ Group( () )^G, Group( [ (1,3)(2,4) ] )^G, Group( [ (3,4) ] )^G,
Group( [ (2,4,3) ] )^G, Group( [ (1,4)(2,3), (1,3)(2,4) ] )^G,
Group( [ (3,4), (1,2)(3,4) ] )^G,
Group( [ (1,3,2,4), (1,2)(3,4) ] )^G, Group( [ (3,4), (2,4,3) ] )^G,
Group( [ (1,4)(2,3), (1,3)(2,4), (3,4) ] )^G,
Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3) ] )^G,
Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ] )^G ]
</pre></div>
<p><a id="X78928A3582882BFD" name="X78928A3582882BFD"></a></p>
<h5>39.20-2 ClassElementLattice</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassElementLattice</code>( <var class="Arg">C</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a class <var class="Arg">C</var> of subgroups, obtained by a lattice computation, this operation returns the <var class="Arg">n</var>-th conjugate subgroup in the class.</p>
<p><em>Because of other methods installed, calling <code class="func">AsList</code> (<a href="chap30_mj.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) with <var class="Arg">C</var> can give a different arrangement of the class elements!</em></p>
<p>The <strong class="pkg">GAP</strong> package <strong class="pkg">XGAP</strong> permits a graphical display of the lattice of subgroups in a nice way.</p>
<p><a id="X7E5DF287825EE7BA" name="X7E5DF287825EE7BA"></a></p>
<h5>39.20-3 DotFileLatticeSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DotFileLatticeSubgroups</code>( <var class="Arg">L</var>, <var class="Arg">file</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function produces a graphical representation of the subgroup lattice <var class="Arg">L</var> in file <var class="Arg">file</var>. The output is in <code class="code">.dot</code> (also known as <code class="code">GraphViz</code> format). For details on the format, and information about how to display or edit this format see <span class="URL"><a href="http://www.graphviz.org">http://www.graphviz.org</a></span>. (On the Macintosh, the program <code class="code">OmniGraffle</code> is also able to read this format.)</p>
<p>Subgroups are labelled in the form <code class="code"><var class="Arg">i</var>-<var class="Arg">j</var></code> where <var class="Arg">i</var> is the number of the class of subgroups and <var class="Arg">j</var> the number within this class. Normal subgroups are represented by a box.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DotFileLatticeSubgroups(l,"s4lat.dot");</span>
</pre></div>
<p><a id="X815CDA447C5DB285" name="X815CDA447C5DB285"></a></p>
<h5>39.20-4 MaximalSubgroupsLattice</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalSubgroupsLattice</code>( <var class="Arg">lat</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a lattice <var class="Arg">lat</var> of subgroups this attribute contains the maximal subgroup relations among the subgroups of the lattice. It is a list corresponding to the <code class="func">ConjugacyClassesSubgroups</code> (<a href="chap39_mj.html#X7E986BF48393113A"><span class="RefLink">39.19-3</span></a>) value of the lattice, each entry giving a list of the maximal subgroups of the representative of this class. Every maximal subgroup is indicated by a list of the form <span class="SimpleMath">\([ c, n ]\)</span> which means that the <span class="SimpleMath">\(n\)</span>-th subgroup in class number <span class="SimpleMath">\(c\)</span> is a maximal subgroup of the representative.</p>
<p>The number <span class="SimpleMath">\(n\)</span> corresponds to access via <code class="func">ClassElementLattice</code> (<a href="chap39_mj.html#X78928A3582882BFD"><span class="RefLink">39.20-2</span></a>) and <em>not</em> necessarily the <code class="func">AsList</code> (<a href="chap30_mj.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) arrangement! See also <code class="func">MinimalSupergroupsLattice</code> (<a href="chap39_mj.html#X8138997C871EDF96"><span class="RefLink">39.20-5</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalSubgroupsLattice(l);</span>
[ [ ], [ [ 1, 1 ] ], [ [ 1, 1 ] ], [ [ 1, 1 ] ],
[ [ 2, 1 ], [ 2, 2 ], [ 2, 3 ] ], [ [ 3, 1 ], [ 3, 6 ], [ 2, 3 ] ],
[ [ 2, 3 ] ], [ [ 4, 1 ], [ 3, 1 ], [ 3, 2 ], [ 3, 3 ] ],
[ [ 7, 1 ], [ 6, 1 ], [ 5, 1 ] ],
[ [ 5, 1 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ], [ 4, 4 ] ],
[ [ 10, 1 ], [ 9, 1 ], [ 9, 2 ], [ 9, 3 ], [ 8, 1 ], [ 8, 2 ],
[ 8, 3 ], [ 8, 4 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">last[6];</span>
[ [ 3, 1 ], [ 3, 6 ], [ 2, 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">u1:=Representative(ConjugacyClassesSubgroups(l)[6]);</span>
Group([ (3,4), (1,2)(3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">u2:=ClassElementLattice(ConjugacyClassesSubgroups(l)[3],1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u3:=ClassElementLattice(ConjugacyClassesSubgroups(l)[3],6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u4:=ClassElementLattice(ConjugacyClassesSubgroups(l)[2],3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(u1,u2);IsSubgroup(u1,u3);IsSubgroup(u1,u4);</span>
true
true
true
</pre></div>
<p><a id="X8138997C871EDF96" name="X8138997C871EDF96"></a></p>
<h5>39.20-5 MinimalSupergroupsLattice</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalSupergroupsLattice</code>( <var class="Arg">lat</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a lattice <var class="Arg">lat</var> of subgroups this attribute contains the minimal supergroup relations among the subgroups of the lattice. It is a list corresponding to the <code class="func">ConjugacyClassesSubgroups</code> (<a href="chap39_mj.html#X7E986BF48393113A"><span class="RefLink">39.19-3</span></a>) value of the lattice, each entry giving a list of the minimal supergroups of the representative of this class. Every minimal supergroup is indicated by a list of the form <span class="SimpleMath">\([ c, n ]\)</span>, which means that the <span class="SimpleMath">\(n\)</span>-th subgroup in class number <span class="SimpleMath">\(c\)</span> is a minimal supergroup of the representative.</p>
<p>The number <span class="SimpleMath">\(n\)</span> corresponds to access via <code class="func">ClassElementLattice</code> (<a href="chap39_mj.html#X78928A3582882BFD"><span class="RefLink">39.20-2</span></a>) and <em>not</em> necessarily the <code class="func">AsList</code> (<a href="chap30_mj.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) arrangement! See also <code class="func">MaximalSubgroupsLattice</code> (<a href="chap39_mj.html#X815CDA447C5DB285"><span class="RefLink">39.20-4</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalSupergroupsLattice(l);</span>
[ [ [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ], [ 3, 2 ], [ 3, 3 ],
[ 3, 4 ], [ 3, 5 ], [ 3, 6 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ],
[ 4, 4 ] ], [ [ 5, 1 ], [ 6, 2 ], [ 7, 2 ] ],
[ [ 6, 1 ], [ 8, 1 ], [ 8, 3 ] ], [ [ 8, 1 ], [ 10, 1 ] ],
[ [ 9, 1 ], [ 9, 2 ], [ 9, 3 ], [ 10, 1 ] ], [ [ 9, 1 ] ],
[ [ 9, 1 ] ], [ [ 11, 1 ] ], [ [ 11, 1 ] ], [ [ 11, 1 ] ], [ ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">last[3];</span>
[ [ 6, 1 ], [ 8, 1 ], [ 8, 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">u5:=ClassElementLattice(ConjugacyClassesSubgroups(l)[8],1);</span>
Group([ (3,4), (2,4,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">u6:=ClassElementLattice(ConjugacyClassesSubgroups(l)[8],3);</span>
Group([ (1,3), (1,3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(u5,u2);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(u6,u2);</span>
true
</pre></div>
<p><a id="X87BE970D7B18E2C5" name="X87BE970D7B18E2C5"></a></p>
<h5>39.20-6 LowLayerSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowLayerSubgroups</code>( <var class="Arg">act</var>, <var class="Arg">G</var>, <var class="Arg">lim</var>, <var class="Arg">cond</var>, <var class="Arg">dosub</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function computes representatives of the conjugacy classes of subgroups of the finite group <var class="Arg">G</var> such that the subgroups can be obtained as <var class="Arg">lim</var>-fold iterated maximal subgroups. If a function <var class="Arg">cond</var> is given, only subgroups for which this function returns true (also for their intermediate overgroups) is returned. If also a function <var class="Arg">dosub</var> is given, maximal subgroups are only attempted if this function returns true (this is separated for performance reasons). In the example below, the result would be the same with leaving out the fourth function, but calculation this way is slightly faster.</p>
<p><a id="X87FABD5F87AD2568" name="X87FABD5F87AD2568"></a></p>
<h5>39.20-7 ContainedConjugates</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ContainedConjugates</code>( <var class="Arg">G</var>, <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For <span class="SimpleMath">\(A,B \leq G\)</span> this operation returns representatives of the <var class="Arg">A</var> conjugacy classes of subgroups that are conjugate to <var class="Arg">B</var> under <var class="Arg">G</var>. The function returns a list of pairs of subgroup and conjugating element.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=TransitiveGroup(8,47);;b:=TransitiveGroup(8,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ContainedConjugates(g,a,b);</span>
[ [ Group([ (1,4,2,5,3,6,8,7), (1,3)(2,8) ]), (2,4,5,3)(7,8) ] ]
</pre></div>
<p><a id="X79C3619C849F97B8" name="X79C3619C849F97B8"></a></p>
<h5>39.20-8 ContainingConjugates</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ContainingConjugates</code>( <var class="Arg">G</var>, <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For <span class="SimpleMath">\(A,B \leq G\)</span> this operation returns all <var class="Arg">G</var> conjugates of <var class="Arg">A</var> that contain <var class="Arg">B</var>. The function returns a list of pairs of subgroup and conjugating element.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=TransitiveGroup(8,47);;b:=TransitiveGroup(8,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ContainingConjugates(g,a,b);</span>
[ [ Group([ (1,3,5,7), (3,5), (1,4)(2,7)(3,6)(5,8) ]), (2,3,5,4)(7,8) ] ]
</pre></div>
<p><a id="X8111F50C798B0D76" name="X8111F50C798B0D76"></a></p>
<h5>39.20-9 MinimalFaithfulPermutationDegree</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalFaithfulPermutationDegree</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a finite group <var class="Arg">G</var> this operation calculates the least positive integer <span class="SimpleMath">\(n=\mu(G)\)</span> such that <var class="Arg">G</var> is isomorphic to a subgroup of the symmetric group of degree <span class="SimpleMath">\(n\)</span>. This can require calculating the whole subgroup lattice.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalFaithfulPermutationDegree(SmallGroup(96,3));</span>
12
</pre></div>
<p><a id="X7BA3484E7AE0A0E1" name="X7BA3484E7AE0A0E1"></a></p>
<h5>39.20-10 RepresentativesPerfectSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesPerfectSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesSimpleSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of conjugacy representatives of perfect (respectively simple) subgroups of <var class="Arg">G</var>. This uses the library of perfect groups (see <code class="func">PerfectGroup</code> (<a href="chap50_mj.html#X7906BBA7818E9415"><span class="RefLink">50.6-2</span></a>)), thus it will issue an error if the library is insufficient to determine all perfect subgroups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m11:=TransitiveGroup(11,6);</span>
M(11)
<span class="GAPprompt">gap></span> <span class="GAPinput">r:=RepresentativesPerfectSubgroups(m11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(r,Size);</span>
[ 60, 60, 360, 660, 7920, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(r,StructureDescription);</span>
[ "A5", "A5", "A6", "PSL(2,11)", "M11", "1" ]
</pre></div>
<p><a id="X7B2233D180DF77A1" name="X7B2233D180DF77A1"></a></p>
<h5>39.20-11 ConjugacyClassesPerfectSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClassesPerfectSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the conjugacy classes of perfect subgroups of <var class="Arg">G</var>. (see <code class="func">RepresentativesPerfectSubgroups</code> (<a href="chap39_mj.html#X7BA3484E7AE0A0E1"><span class="RefLink">39.20-10</span></a>).)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r := ConjugacyClassesPerfectSubgroups(m11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(r, x -> StructureDescription(Representative(x)));</span>
[ "A5", "A5", "A6", "PSL(2,11)", "M11", "1" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SortedList( List(r,Size) );</span>
[ 1, 1, 11, 12, 66, 132 ]
</pre></div>
<p><a id="X7BFE573187B4BEF8" name="X7BFE573187B4BEF8"></a></p>
<h5>39.20-12 Zuppos</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Zuppos</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <em>Zuppos</em> of a group are the cyclic subgroups of prime power order. (The name "Zuppo" derives from the German abbreviation for "zyklische Untergruppen von Primzahlpotenzordnung".) This attribute gives generators of all such subgroups of a group <var class="Arg">G</var>. That is all elements of <var class="Arg">G</var> of prime power order up to the equivalence that they generate the same cyclic subgroup.</p>
<p><a id="X82C12E2C81963B23" name="X82C12E2C81963B23"></a></p>
<h5>39.20-13 InfoLattice</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoLattice</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>is the information class used by the cyclic extension methods for subgroup lattice calculations.</p>
<p><a id="X85E613D57F28AEFF" name="X85E613D57F28AEFF"></a></p>
<h4>39.21 <span class="Heading">Specific Methods for Subgroup Lattice Computations</span></h4>
<p><a id="X86462A567DDBA6BC" name="X86462A567DDBA6BC"></a></p>
<h5>39.21-1 LatticeByCyclicExtension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LatticeByCyclicExtension</code>( <var class="Arg">G</var>[, <var class="Arg">func</var>[, <var class="Arg">noperf</var>]] )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the lattice of <var class="Arg">G</var> using the cyclic extension algorithm. If the function <var class="Arg">func</var> is given, the algorithm will discard all subgroups not fulfilling <var class="Arg">func</var> (and will also not extend them), returning a partial lattice. This can be useful to compute only subgroups with certain properties. Note however that this will <em>not</em> necessarily yield all subgroups that fulfill <var class="Arg">func</var>, but the subgroups whose subgroups are used for the construction must also fulfill <var class="Arg">func</var> as well. (In fact the filter <var class="Arg">func</var> will simply discard subgroups in the cyclic extension algorithm. Therefore the trivial subgroup will always be included.) Also note, that for such a partial lattice maximality/minimality inclusion relations cannot be computed. (If <var class="Arg">func</var> is a list of length 2, its first entry is such a discarding function, the second a function for discarding zuppos.)</p>
<p>The cyclic extension algorithm requires the perfect subgroups of <var class="Arg">G</var>. However <strong class="pkg">GAP</strong> cannot analyze the function <var class="Arg">func</var> for its implication but can only apply it. If it is known that <var class="Arg">func</var> implies solvability, the computation of the perfect subgroups can be avoided by giving a third parameter <var class="Arg">noperf</var> set to <code class="keyw">true</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=WreathProduct(Group((1,2,3),(1,2)),Group((1,2,3,4)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=LatticeByCyclicExtension(g,function(G)</span>
<span class="GAPprompt">></span> <span class="GAPinput">return Size(G) in [1,2,3,6];end);</span>
<subgroup lattice of <permutation group of size 5184 with
9 generators>, 47 classes,
2628 subgroups, restricted under further condition l!.func>
</pre></div>
<p>The total number of classes in this example is much bigger, as the following example shows:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LatticeSubgroups(g);</span>
<subgroup lattice of <permutation group of size 5184 with
9 generators>, 566 classes, 27134 subgroups>
</pre></div>
<p>##</p>
<p><a id="X78918D83835A0EDF" name="X78918D83835A0EDF"></a></p>
<h5>39.21-2 InvariantSubgroupsElementaryAbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InvariantSubgroupsElementaryAbelianGroup</code>( <var class="Arg">G</var>, <var class="Arg">homs</var>[, <var class="Arg">dims</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be an elementary abelian group and <var class="Arg">homs</var> be a set of automorphisms of <var class="Arg">G</var>. Then this function computes all subspaces of <var class="Arg">G</var> which are invariant under all automorphisms in <var class="Arg">homs</var>. When considering <var class="Arg">G</var> as a module for the algebra generated by <var class="Arg">homs</var>, these are all submodules. If <var class="Arg">homs</var> is empty, it computes all subgroups. If the optional parameter <var class="Arg">dims</var> is given, only submodules of this dimension are computed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3),(4,5,6),(7,8,9));</span>
Group([ (1,2,3), (4,5,6), (7,8,9) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=GroupHomomorphismByImages(g,g,[(1,2,3),(4,5,6),(7,8,9)],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[(7,8,9),(1,2,3),(4,5,6)]);</span>
[ (1,2,3), (4,5,6), (7,8,9) ] -> [ (7,8,9), (1,2,3), (4,5,6) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">u:=InvariantSubgroupsElementaryAbelianGroup(g,[hom]);</span>
[ Group(()), Group([ (1,2,3)(4,5,6)(7,8,9) ]),
Group([ (1,3,2)(7,8,9), (1,3,2)(4,5,6) ]),
Group([ (7,8,9), (4,5,6), (1,2,3) ]) ]
</pre></div>
<p><a id="X7AD7804A803910AC" name="X7AD7804A803910AC"></a></p>
<h5>39.21-3 SubgroupsSolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubgroupsSolvableGroup</code>( <var class="Arg">G</var>[, <var class="Arg">opt</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function (implementing the algorithm published in <a href="chapBib_mj.html#biBHulpke99">[Hul99]</a>) computes subgroups of a solvable group <var class="Arg">G</var>, using the homomorphism principle. It returns a list of representatives up to <var class="Arg">G</var>-conjugacy.</p>
<p>The optional argument <var class="Arg">opt</var> is a record, which may be used to suggest restrictions on the subgroups computed. The following record components of <var class="Arg">opt</var> are recognized and have the following effects. Note that all of the following restrictions to subgroups with particular properties are only used to speed up the calculation, but the result might still contain subgroups (that had to be computed in any case) that do not satisfy the properties. If this is not desired, the calculation must be followed by an explicit test for the desired properties (which is not done by default, as it would be a general slowdown). The function guarantees that representatives of all subgroups that satisfy the properties are found, i.e. there can be only false positives.</p>
<dl>
<dt><strong class="Mark"><code class="code">actions</code></strong></dt>
<dd><p>must be a list of automorphisms of <var class="Arg">G</var>. If given, only groups which are invariant under all these automorphisms are computed. The algorithm must know the normalizer in <var class="Arg">G</var> of the group generated by <code class="code">actions</code> (defined formally by embedding in the semidirect product of <var class="Arg">G</var> with <var class="Arg">actions</var>). This can be given in the component <code class="code">funcnorm</code> and will be computed if this component is not given.</p>
</dd>
<dt><strong class="Mark"><code class="code">normal</code></strong></dt>
<dd><p>if set to <code class="keyw">true</code> only normal subgroups are guaranteed to be returned (though some of the returned subgroups might still be not normal).</p>
</dd>
<dt><strong class="Mark"><code class="code">consider</code></strong></dt>
<dd><p>a function to restrict the groups computed. This must be a function of five parameters, <span class="SimpleMath">\(C\)</span>, <span class="SimpleMath">\(A\)</span>, <span class="SimpleMath">\(N\)</span>, <span class="SimpleMath">\(B\)</span>, <span class="SimpleMath">\(M\)</span>, that are interpreted as follows: The arguments are subgroups of a factor <span class="SimpleMath">\(F\)</span> of <var class="Arg">G</var> in the relation <span class="SimpleMath">\(F \geq C > A > N > B > M\)</span>. <span class="SimpleMath">\(N\)</span> and <span class="SimpleMath">\(M\)</span> are normal subgroups. <span class="SimpleMath">\(C\)</span> is the full preimage of the normalizer of <span class="SimpleMath">\(A/N\)</span> in <span class="SimpleMath">\(F/N\)</span>. When computing modulo <span class="SimpleMath">\(M\)</span> and looking for subgroups <span class="SimpleMath">\(U\)</span> such that <span class="SimpleMath">\(U \cap N = B\)</span> and <span class="SimpleMath">\(\langle U, N \rangle = A\)</span>, this function is called. If it returns <code class="keyw">false</code> then all potential groups <span class="SimpleMath">\(U\)</span> (and therefore all groups later arising from them) are disregarded. This can be used for example to compute only subgroups of certain sizes.</p>
<p>(<em>This is just a restriction to speed up computations. The function may still return (invariant) subgroups which don't fulfill this condition!</em>) This parameter is used to permit calculations of some subgroups if the set of all subgroups would be too large to handle.</p>
<p>The actual groups <span class="SimpleMath">\(C\)</span>, <span class="SimpleMath">\(A\)</span>, <span class="SimpleMath">\(N\)</span> and <span class="SimpleMath">\(B\)</span> which are passed to this function are not necessarily subgroups of <var class="Arg">G</var> but might be subgroups of a proper factor group <span class="SimpleMath">\(F = \textit{G}/H\)</span>. Therefore the <code class="code">consider</code> function may not relate the parameter groups to <var class="Arg">G</var>.</p>
</dd>
<dt><strong class="Mark"><code class="code">retnorm</code></strong></dt>
<dd><p>if set to <code class="keyw">true</code> the function not only returns a list <code class="code">subs</code> of subgroups but also a corresponding list <code class="code">norms</code> of normalizers in the form <code class="code">[ subs, norms ]</code>.</p>
</dd>
<dt><strong class="Mark"><code class="code">series</code></strong></dt>
<dd><p>is an elementary abelian series of <var class="Arg">G</var> which will be used for the computation.</p>
</dd>
<dt><strong class="Mark"><code class="code">groups</code></strong></dt>
<dd><p>is a list of groups to seed the calculation. Only subgroups of these groups are constructed.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3),(1,2),(4,5,6),(4,5),(7,8,9),(7,8));</span>
Group([ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=GroupHomomorphismByImages(g,g,</span>
<span class="GAPprompt">></span> <span class="GAPinput">[(1,2,3),(1,2),(4,5,6),(4,5),(7,8,9),(7,8)],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[(4,5,6),(4,5),(7,8,9),(7,8),(1,2,3),(1,2)]);</span>
[ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8) ] ->
[ (4,5,6), (4,5), (7,8,9), (7,8), (1,2,3), (1,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=SubgroupsSolvableGroup(g,rec(actions:=[hom]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(l,Size);</span>
[ 1, 3, 9, 27, 54, 2, 6, 18, 108, 4, 216, 8 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(ConjugacyClassesSubgroups(g)); # to compare</span>
162
</pre></div>
<p><a id="X7F60BBB8874DFE40" name="X7F60BBB8874DFE40"></a></p>
<h5>39.21-4 SizeConsiderFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SizeConsiderFunction</code>( <var class="Arg">size</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a function <code class="code">consider</code> of four arguments that can be used in <code class="func">SubgroupsSolvableGroup</code> (<a href="chap39_mj.html#X7AD7804A803910AC"><span class="RefLink">39.21-3</span></a>) for the option <code class="code">consider</code> to compute subgroups whose sizes are divisible by <var class="Arg">size</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=SubgroupsSolvableGroup(g,rec(actions:=[hom],</span>
<span class="GAPprompt">></span> <span class="GAPinput">consider:=SizeConsiderFunction(6)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(l,Size);</span>
[ 1, 3, 9, 27, 54, 6, 18, 108, 216 ]
</pre></div>
<p>This example shows that in general the <code class="code">consider</code> function does not provide a perfect filter. It is guaranteed that all subgroups fulfilling the condition are returned, but not all subgroups returned necessarily fulfill the condition.</p>
<p><a id="X833C51BD7E7812C4" name="X833C51BD7E7812C4"></a></p>
<h5>39.21-5 ExactSizeConsiderFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExactSizeConsiderFunction</code>( <var class="Arg">size</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a function <code class="code">consider</code> of four arguments that can be used in <code class="func">SubgroupsSolvableGroup</code> (<a href="chap39_mj.html#X7AD7804A803910AC"><span class="RefLink">39.21-3</span></a>) for the option <code class="code">consider</code> to compute subgroups whose sizes are exactly <var class="Arg">size</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=SubgroupsSolvableGroup(g,rec(actions:=[hom],</span>
<span class="GAPprompt">></span> <span class="GAPinput">consider:=ExactSizeConsiderFunction(6)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(l,Size);</span>
[ 1, 3, 9, 27, 54, 6, 108, 216 ]
</pre></div>
<p>Again, the <code class="code">consider</code> function does not provide a perfect filter. It is guaranteed that all subgroups fulfilling the condition are returned, but not all subgroups returned necessarily fulfill the condition.</p>
<p><a id="X7A2C774B7CFF3E07" name="X7A2C774B7CFF3E07"></a></p>
<h5>39.21-6 InfoPcSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoPcSubgroup</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>Information function for the subgroup lattice functions using pcgs.</p>
<p><a id="X79F894537D526B61" name="X79F894537D526B61"></a></p>
<h4>39.22 <span class="Heading">Special Generating Sets</span></h4>
<p><a id="X82FD78AF7F80A0E2" name="X82FD78AF7F80A0E2"></a></p>
<h5>39.22-1 GeneratorsSmallest</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsSmallest</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a "smallest" generating set for the group <var class="Arg">G</var>. This is the lexicographically (using <strong class="pkg">GAP</strong>s order of group elements) smallest list <span class="SimpleMath">\(l\)</span> of elements of <var class="Arg">G</var> such that <span class="SimpleMath">\(G = \langle l \rangle\)</span> and <span class="SimpleMath">\(l_i \not \in \langle l_1, \ldots, l_{{i-1}} \rangle\)</span> (in particular <span class="SimpleMath">\(l_1\)</span> is not the identity element of the group). The comparison of two groups via lexicographic comparison of their sorted element lists yields the same relation as lexicographic comparison of their smallest generating sets.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsSmallest(g);</span>
[ (3,4), (2,3), (1,2) ]
</pre></div>
<p><a id="X7A258CCF79552198" name="X7A258CCF79552198"></a></p>
<h5>39.22-2 LargestElementGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LargestElementGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the largest element of <var class="Arg">G</var> with respect to the ordering <code class="code"><</code> of the elements family.</p>
<p><a id="X81D15723804771E2" name="X81D15723804771E2"></a></p>
<h5>39.22-3 MinimalGeneratingSet</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalGeneratingSet</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a generating set of <var class="Arg">G</var> of minimal possible length.</p>
<p>Note that –apart from special cases– currently there are only efficient methods known to compute minimal generating sets of finite solvable groups and of finitely generated nilpotent groups. Hence so far these are the only cases for which methods are available. The former case is covered by a method implemented in the <strong class="pkg">GAP</strong> library, while the second case requires the package <strong class="pkg">Polycyclic</strong>.</p>
<p>If you do not really need a minimal generating set, but are satisfied with getting a reasonably small set of generators, you better use <code class="func">SmallGeneratingSet</code> (<a href="chap39_mj.html#X814DBABC878D5232"><span class="RefLink">39.22-4</span></a>).</p>
<p>Information about the minimal generating sets of the finite simple groups of order less than <span class="SimpleMath">\(10^6\)</span> can be found in <a href="chapBib_mj.html#biBMY79">[MY79]</a>. See also the package <strong class="pkg">AtlasRep</strong>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGeneratingSet(g);</span>
[ (2,4,3), (1,4,2,3) ]
</pre></div>
<p><a id="X814DBABC878D5232" name="X814DBABC878D5232"></a></p>
<h5>39.22-4 SmallGeneratingSet</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallGeneratingSet</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a generating set of <var class="Arg">G</var> which has few elements. As neither irredundancy, nor minimal length is proven it runs much faster than <code class="func">MinimalGeneratingSet</code> (<a href="chap39_mj.html#X81D15723804771E2"><span class="RefLink">39.22-3</span></a>). It can be used whenever a short generating set is desired which not necessarily needs to be optimal.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallGeneratingSet(g);</span>
[ (1,2,3,4), (1,2) ]
</pre></div>
<p><a id="X7D1574457B152333" name="X7D1574457B152333"></a></p>
<h5>39.22-5 IndependentGeneratorsOfAbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IndependentGeneratorsOfAbelianGroup</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of generators <span class="SimpleMath">\(a_1, a_2, \ldots\)</span> of prime power order or infinite order of the abelian group <var class="Arg">A</var> such that <var class="Arg">A</var> is the direct product of the cyclic groups generated by the <span class="SimpleMath">\(a_i\)</span>. The list of orders of the returned generators must match the result of <code class="func">AbelianInvariants</code> (<a href="chap39_mj.html#X812827937F403300"><span class="RefLink">39.16-1</span></a>) (taking into account that zero and <code class="func">infinity</code> (<a href="chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) are identified).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=AbelianGroup(IsPermGroup,[15,14,22,78]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(IndependentGeneratorsOfAbelianGroup(g),Order);</span>
[ 2, 2, 2, 3, 3, 5, 7, 11, 13 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(g);</span>
[ 2, 2, 2, 3, 3, 5, 7, 11, 13 ]
</pre></div>
<p><a id="X86F835DA8264A0CE" name="X86F835DA8264A0CE"></a></p>
<h5>39.22-6 IndependentGeneratorExponents</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IndependentGeneratorExponents</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For an abelian group <var class="Arg">G</var>, with <code class="func">IndependentGeneratorsOfAbelianGroup</code> (<a href="chap39_mj.html#X7D1574457B152333"><span class="RefLink">39.22-5</span></a>) value the list <span class="SimpleMath">\([ a_1, \ldots, a_n ]\)</span>, this operation returns the exponent vector <span class="SimpleMath">\([ e_1, \ldots, e_n ]\)</span> to represent <span class="SimpleMath">\(\textit{g} = \prod_i a_i^{{e_i}}\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := AbelianGroup([16,9,625]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := IndependentGeneratorsOfAbelianGroup(g);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(gens, Order);</span>
[ 9, 16, 625 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(g);</span>
[ 9, 16, 625 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">r:=gens[1]^4*gens[2]^12*gens[3]^128;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IndependentGeneratorExponents(g,r);</span>
[ 4, 12, 128 ]
</pre></div>
<p><a id="X7CA0B6A27E0BE6B8" name="X7CA0B6A27E0BE6B8"></a></p>
<h4>39.23 <span class="Heading">1-Cohomology</span></h4>
<p>Let <span class="SimpleMath">\(G\)</span> be a finite group and <span class="SimpleMath">\(M\)</span> an elementary abelian normal <span class="SimpleMath">\(p\)</span>-subgroup of <span class="SimpleMath">\(G\)</span>. Then the group of 1-cocycles <span class="SimpleMath">\(Z^1( G/M, M )\)</span> is defined as</p>
<p class="center">\[
Z^1(G/M, M) = \{ \gamma: G/M \rightarrow M \mid \forall g_1, g_2 \in G :
\gamma(g_1 M \cdot g_2 M )
= \gamma(g_1 M)^{{g_2}} \cdot \gamma(g_2 M) \}
\]</p>
<p>and is a <span class="SimpleMath">\(GF(p)\)</span>-vector space.</p>
<p>The group of 1-coboundaries <span class="SimpleMath">\(B^1( G/M, M )\)</span> is defined as</p>
<p class="center">\[
B^1(G/M, M) = \{ \gamma : G/M \rightarrow M \mid \exists m \in M
\forall g \in G :
\gamma(gM) = (m^{{-1}})^g \cdot m \}
\]</p>
<p>It also is a <span class="SimpleMath">\(GF(p)\)</span>-vector space.</p>
<p>Let <span class="SimpleMath">\(\alpha\)</span> be the isomorphism of <span class="SimpleMath">\(M\)</span> into a row vector space <span class="SimpleMath">\({\cal W}\)</span> and <span class="SimpleMath">\((g_1, \ldots, g_l)\)</span> representatives for a generating set of <span class="SimpleMath">\(G/M\)</span>. Then there exists a monomorphism <span class="SimpleMath">\(\beta\)</span> of <span class="SimpleMath">\(Z^1( G/M, M )\)</span> in the <span class="SimpleMath">\(l\)</span>-fold direct sum of <span class="SimpleMath">\({\cal W}\)</span>, such that <span class="SimpleMath">\(\beta( \gamma ) = ( \alpha( \gamma(g_1 M) ),\ldots, \alpha( \gamma(g_l M) ) )\)</span> for every <span class="SimpleMath">\(\gamma \in Z^1( G/M, M )\)</span>.</p>
<p><a id="X847BEC137A49BAF4" name="X847BEC137A49BAF4"></a></p>
<h5>39.23-1 <span class="Heading">OneCocycles</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneCocycles</code>( <var class="Arg">G</var>, <var class="Arg">M</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">‣ OneCocycles</code>( <var class="Arg">G</var>, <var class="Arg">mpcgs</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">‣ OneCocycles</code>( <var class="Arg">gens</var>, <var class="Arg">M</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">‣ OneCocycles</code>( <var class="Arg">gens</var>, <var class="Arg">mpcgs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Computes the group of 1-cocycles <span class="SimpleMath">\(Z^1(\textit{G}/\textit{M},\textit{M})\)</span>. The normal subgroup <var class="Arg">M</var> may be given by a (Modulo)Pcgs <var class="Arg">mpcgs</var>. In this case the whole calculation is performed modulo the normal subgroup defined by <code class="code">DenominatorOfModuloPcgs(<var class="Arg">mpcgs</var>)</code> (see <a href="chap45_mj.html#X7F18A01785DBAC4E"><span class="RefLink">45.1</span></a>). Similarly the group <var class="Arg">G</var> may instead be specified by a set of elements <var class="Arg">gens</var> that are representatives for a generating system for the factor group <var class="Arg">G</var>/<var class="Arg">M</var>. If this is done the 1-cocycles are computed with respect to these generators (otherwise the routines try to select suitable generators themselves). The current version of the code assumes that <var class="Arg">G</var> is a permutation group or a pc group.</p>
<p><a id="X7E6438D5834ACCDA" name="X7E6438D5834ACCDA"></a></p>
<h5>39.23-2 OneCoboundaries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneCoboundaries</code>( <var class="Arg">G</var>, <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the group of 1-coboundaries. Syntax of input and output otherwise is the same as with <code class="func">OneCocycles</code> (<a href="chap39_mj.html#X847BEC137A49BAF4"><span class="RefLink">39.23-1</span></a>) except that entries that refer to cocycles are not computed.</p>
<p>The operations <code class="func">OneCocycles</code> (<a href="chap39_mj.html#X847BEC137A49BAF4"><span class="RefLink">39.23-1</span></a>) and <code class="func">OneCoboundaries</code> return a record with (at least) the components:</p>
<dl>
<dt><strong class="Mark"><code class="code">generators</code></strong></dt>
<dd><p>Is a list of representatives for a generating set of <var class="Arg">G</var>/<var class="Arg">M</var>. Cocycles are represented with respect to these generators.</p>
</dd>
<dt><strong class="Mark"><code class="code">oneCocycles</code></strong></dt>
<dd><p>A space of row vectors over GF(<span class="SimpleMath">\(p\)</span>), representing <span class="SimpleMath">\(Z^1\)</span>. The vectors are represented in dimension <span class="SimpleMath">\(a \cdot b\)</span> where <span class="SimpleMath">\(a\)</span> is the length of <code class="code">generators</code> and <span class="SimpleMath">\(p^b\)</span> the size of <var class="Arg">M</var>.</p>
</dd>
<dt><strong class="Mark"><code class="code">oneCoboundaries</code></strong></dt>
<dd><p>A space of row vectors that represents <span class="SimpleMath">\(B^1\)</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">cocycleToList</code></strong></dt>
<dd><p>is a function to convert a cocycle (a row vector in <code class="code">oneCocycles</code>) to a corresponding list of elements of <var class="Arg">M</var>.</p>
</dd>
<dt><strong class="Mark"><code class="code">listToCocycle</code></strong></dt>
<dd><p>is a function to convert a list of elements of <var class="Arg">M</var> to a cocycle.</p>
</dd>
<dt><strong class="Mark"><code class="code">isSplitExtension</code></strong></dt>
<dd><p>indicates whether <var class="Arg">G</var> splits over <var class="Arg">M</var>. The following components are only bound if the extension splits. Note that if <var class="Arg">M</var> is given by a modulo pcgs all subgroups are given as subgroups of <var class="Arg">G</var> by generators corresponding to <code class="code">generators</code> and thus may not contain the denominator of the modulo pcgs. In this case taking the closure with this denominator will give the full preimage of the complement in the factor group.</p>
</dd>
<dt><strong class="Mark"><code class="code">complement</code></strong></dt>
<dd><p>One complement to <var class="Arg">M</var> in <var class="Arg">G</var>.</p>
</dd>
<dt><strong class="Mark"><code class="code">cocycleToComplement( cyc )</code></strong></dt>
<dd><p>is a function that takes a cocycle from <code class="code">oneCocycles</code> and returns the corresponding complement to <var class="Arg">M</var> in <var class="Arg">G</var> (with respect to the fixed complement <code class="code">complement</code>).</p>
</dd>
<dt><strong class="Mark"><code class="code">complementToCocycle(<var class="Arg">U</var>)</code></strong></dt>
<dd><p>is a function that takes a complement and returns the corresponding cocycle.</p>
</dd>
</dl>
<p>If the factor <var class="Arg">G</var>/<var class="Arg">M</var> is given by a (modulo) pcgs <var class="Arg">gens</var> then special methods are used that compute a presentation for the factor implicitly from the pcgs.</p>
<p>Note that the groups of 1-cocycles and 1-coboundaries are not groups in the sense of <code class="func">Group</code> (<a href="chap39_mj.html#X7D8E473384DE9CD4"><span class="RefLink">39.2-1</span></a>) for <strong class="pkg">GAP</strong> but vector spaces.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=Group((1,2)(3,4),(1,3)(2,4));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">oc:=OneCocycles(g,n);</span>
rec( cocycleToComplement := function( c ) ... end,
cocycleToList := function( c ) ... end,
complement := Group([ (3,4), (2,4,3) ]),
complementGens := [ (3,4), (2,4,3) ],
complementToCocycle := function( K ) ... end,
factorGens := [ (3,4), (2,4,3) ], generators := [ (3,4), (2,4,3) ],
isSplitExtension := true, listToCocycle := function( L ) ... end,
oneCoboundaries := <vector space over GF(2), with 2 generators>,
oneCocycles := <vector space over GF(2), with 2 generators> )
<span class="GAPprompt">gap></span> <span class="GAPinput">oc.cocycleToList([ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ]);</span>
[ (1,2)(3,4), (1,2)(3,4) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">oc.listToCocycle([(),(1,3)(2,4)]) = Z(2) * [ 0, 0, 1, 0];</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">oc.cocycleToComplement([ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ]);</span>
Group([ (3,4), (1,3,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">oc.complementToCocycle(Group((1,2,4),(1,4))) = Z(2) * [ 0, 1, 1, 1 ];</span>
true
</pre></div>
<p>The factor group <span class="SimpleMath">\(H^1(\textit{G}/\textit{M}, \textit{M}) = Z^1(\textit{G}/\textit{M}, \textit{M}) / B^1(\textit{G}/\textit{M}, \textit{M})\)</span> is called the first cohomology group. Currently there is no function which explicitly computes this group. The easiest way to represent it is as a vector space complement to <span class="SimpleMath">\(B^1\)</span> in <span class="SimpleMath">\(Z^1\)</span>.</p>
<p>If the only purpose of the calculation of <span class="SimpleMath">\(H^1\)</span> is the determination of complements it might be desirable to stop calculations once it is known that the extension cannot split. This can be achieved via the more technical function <code class="func">OCOneCocycles</code> (<a href="chap39_mj.html#X80400ABD7F40FAA0"><span class="RefLink">39.23-3</span></a>).</p>
<p><a id="X80400ABD7F40FAA0" name="X80400ABD7F40FAA0"></a></p>
<h5>39.23-3 OCOneCocycles</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OCOneCocycles</code>( <var class="Arg">ocr</var>, <var class="Arg">onlySplit</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the more technical function to compute 1-cocycles. It takes a record <var class="Arg">ocr</var> as first argument which must contain at least the components <code class="code">group</code> for the group and <code class="code">modulePcgs</code> for a (modulo) pcgs of the module. This record will also be returned with components as described under <code class="func">OneCocycles</code> (<a href="chap39_mj.html#X847BEC137A49BAF4"><span class="RefLink">39.23-1</span></a>) (with the exception of <code class="code">isSplitExtension</code> which is indicated by the existence of a <code class="code">complement</code>) but components such as <code class="code">oneCoboundaries</code> will only be computed if not already present.</p>
<p>If <var class="Arg">onlySplit</var> is <code class="keyw">true</code>, <code class="func">OCOneCocycles</code> returns <code class="keyw">false</code> as soon as possible if the extension does not split.</p>
<p><a id="X811E1CF07DABE924" name="X811E1CF07DABE924"></a></p>
<h5>39.23-4 ComplementClassesRepresentativesEA</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComplementClassesRepresentativesEA</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes complement classes to an elementary abelian normal subgroup <var class="Arg">N</var> via 1-Cohomology. Normally, a user program should call <code class="func">ComplementClassesRepresentatives</code> (<a href="chap39_mj.html#X811B8A4683DDE1F9"><span class="RefLink">39.11-6</span></a>) instead, which also works for a solvable (not necessarily elementary abelian) <var class="Arg">N</var>.</p>
<p><a id="X8199B1D27D487897" name="X8199B1D27D487897"></a></p>
<h5>39.23-5 InfoCoh</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoCoh</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>The info class for the cohomology calculations is <code class="func">InfoCoh</code>.</p>
<p><a id="X80A4B0F282977074" name="X80A4B0F282977074"></a></p>
<h4>39.24 <span class="Heading">Schur Covers and Multipliers</span></h4>
<p>Additional attributes and properties of a group can be derived from computing its Schur cover. For example, if <span class="SimpleMath">\(G\)</span> is a finitely presented group, the derived subgroup of a Schur cover of <span class="SimpleMath">\(G\)</span> is invariant and isomorphic to the <code class="func">NonabelianExteriorSquare</code> (<a href="chap39_mj.html#X8739CD4686301A0E"><span class="RefLink">39.24-5</span></a>) value of <span class="SimpleMath">\(G\)</span>, see <a href="chapBib_mj.html#biBBJR87">[BJR87]</a>.</p>
<p><a id="X7F619DDA7DD6C43B" name="X7F619DDA7DD6C43B"></a></p>
<h5>39.24-1 EpimorphismSchurCover</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EpimorphismSchurCover</code>( <var class="Arg">G</var>[, <var class="Arg">pl</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an epimorphism <span class="SimpleMath">\(epi\)</span> from a group <span class="SimpleMath">\(D\)</span> onto <var class="Arg">G</var>. The group <span class="SimpleMath">\(D\)</span> is one (of possibly several) Schur covers of <var class="Arg">G</var>. The group <span class="SimpleMath">\(D\)</span> can be obtained as the <code class="func">Source</code> (<a href="chap32_mj.html#X7DE8173F80E07AB1"><span class="RefLink">32.3-8</span></a>) value of <var class="Arg">epi</var>. The kernel of <span class="SimpleMath">\(epi\)</span> is the Schur multiplier of <var class="Arg">G</var>. If <var class="Arg">pl</var> is given as a list of primes, only the multiplier part for these primes is realized. At the moment, <span class="SimpleMath">\(D\)</span> is represented as a finitely presented group.</p>
<p><a id="X7DD1E37987612042" name="X7DD1E37987612042"></a></p>
<h5>39.24-2 SchurCover</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SchurCover</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns one (of possibly several) Schur covers of the group <var class="Arg">G</var>.</p>
<p>At the moment this cover is represented as a finitely presented group and <code class="func">IsomorphismPermGroup</code> (<a href="chap43_mj.html#X80B7B1C783AA1567"><span class="RefLink">43.3-1</span></a>) would be needed to convert it to a permutation group.</p>
<p>If also the relation to <var class="Arg">G</var> is needed, <code class="func">EpimorphismSchurCover</code> (<a href="chap39_mj.html#X7F619DDA7DD6C43B"><span class="RefLink">39.24-1</span></a>) should be used.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">epi:=EpimorphismSchurCover(g);</span>
[ f1, f2, f3 ] -> [ (3,4), (2,4,3), (1,3)(2,4) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(Source(epi));</span>
48
</pre></div>
<p>If the group becomes bigger, Schur Cover calculations might become unfeasible.</p>
<p>There is another operation, <code class="func">AbelianInvariantsMultiplier</code> (<a href="chap39_mj.html#X792BC39D7CEB1D27"><span class="RefLink">39.24-3</span></a>), which only returns the structure of the Schur Multiplier, and which should work for larger groups as well.</p>
<p><a id="X792BC39D7CEB1D27" name="X792BC39D7CEB1D27"></a></p>
<h5>39.24-3 AbelianInvariantsMultiplier</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianInvariantsMultiplier</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the abelian invariants of the Schur multiplier of <var class="Arg">G</var>.</p>
<p>At the moment, this operation will not give any information about how to extend the multiplier to a Schur Cover.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariantsMultiplier(g);</span>
[ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariantsMultiplier(AlternatingGroup(6));</span>
[ 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariantsMultiplier(SL(2,3));</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariantsMultiplier(SL(3,2));</span>
[ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariantsMultiplier(PSU(4,2));</span>
[ 2 ]
</pre></div>
<p>(Note that the last command from the example will take some time.)</p>
<p>The <strong class="pkg">GAP</strong> 4.4.12 manual contained examples for larger groups e.g. <span class="SimpleMath">\(M_{22}\)</span>. However, some issues that may very rarely (and not easily reproducibly) lead to wrong results were discovered in the code capable of handling larger groups, and in <strong class="pkg">GAP</strong> 4.5 it was replaced by a more reliable basic method. To deal with larger groups, one can use the function <code class="func">SchurMultiplier</code> (<span class="RefLink">???</span>) from the <strong class="pkg">cohomolo</strong> package. Also, additional methods for <code class="func">AbelianInvariantsMultiplier</code> are installed in the <strong class="pkg">Polycyclic</strong> package for pcp-groups.</p>
<p><a id="X819E8AEC835F8CD1" name="X819E8AEC835F8CD1"></a></p>
<h5>39.24-4 Epicentre</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Epicentre</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExteriorCentre</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>There are various ways of describing the epicentre of a group <var class="Arg">G</var>. It is the smallest normal subgroup <span class="SimpleMath">\(N\)</span> of <var class="Arg">G</var> such that <span class="SimpleMath">\(\textit{G}/N\)</span> is a central quotient of a group. It is also equal to the Exterior Center of <var class="Arg">G</var>, see <a href="chapBib_mj.html#biBEllis98">[Ell98]</a>.</p>
<p><a id="X8739CD4686301A0E" name="X8739CD4686301A0E"></a></p>
<h5>39.24-5 NonabelianExteriorSquare</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonabelianExteriorSquare</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Computes the nonabelian exterior square <span class="SimpleMath">\(\textit{G} \wedge \textit{G}\)</span> of the group <var class="Arg">G</var>, which for a finitely presented group is the derived subgroup of any Schur cover of <var class="Arg">G</var> (see <a href="chapBib_mj.html#biBBJR87">[BJR87]</a>).</p>
<p><a id="X7E1C8CD77CDB9F71" name="X7E1C8CD77CDB9F71"></a></p>
<h5>39.24-6 EpimorphismNonabelianExteriorSquare</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EpimorphismNonabelianExteriorSquare</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Computes the mapping <span class="SimpleMath">\(\textit{G} \wedge \textit{G} \rightarrow \textit{G}\)</span>. The kernel of this mapping is equal to the Schur multiplier of <var class="Arg">G</var>.</p>
<p><a id="X7BF8DB3D8300BB3F" name="X7BF8DB3D8300BB3F"></a></p>
<h5>39.24-7 IsCentralFactor</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCentralFactor</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This function determines if there exists a group <span class="SimpleMath">\(H\)</span> such that <var class="Arg">G</var> is isomorphic to the quotient <span class="SimpleMath">\(H/Z(H)\)</span>. A group with this property is called in literature <em>capable</em>. A group being capable is equivalent to the epicentre of <var class="Arg">G</var> being trivial, see <a href="chapBib_mj.html#biBBFS79">[BFS79]</a>.</p>
<p><a id="X7F4240CD782B6032" name="X7F4240CD782B6032"></a></p>
<h5>39.24-8 <span class="Heading">Covering groups of symmetric groups</span></h5>
<p>The covering groups of symmetric groups were classified in <a href="chapBib_mj.html#biBSchur1911">[Sch11]</a>; an inductive procedure to construct faithful, irreducible representations of minimal degree over all fields was presented in <a href="chapBib_mj.html#biBMaas2010">[Maa10]</a>. Methods for <code class="func">EpimorphismSchurCover</code> (<a href="chap39_mj.html#X7F619DDA7DD6C43B"><span class="RefLink">39.24-1</span></a>) are provided for natural symmetric groups which use these representations. For alternating groups, the restriction of these representations are provided, but they may not be irreducible. In the case of degree <span class="SimpleMath">\(6\)</span> and <span class="SimpleMath">\(7\)</span>, they are not the full covering groups and so matrix representations are just stored explicitly for the six-fold covers.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">EpimorphismSchurCover(SymmetricGroup(15));</span>
[ < immutable compressed matrix 64x64 over GF(9) >,
< immutable compressed matrix 64x64 over GF(9) > ] ->
[ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (1,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">EpimorphismSchurCover(AlternatingGroup(15));</span>
[ < immutable compressed matrix 64x64 over GF(9) >,
< immutable compressed matrix 64x64 over GF(9) > ] ->
[ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (13,14,15) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SchurCoverOfSymmetricGroup(12);</span>
<matrix group of size 958003200 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">DoubleCoverOfAlternatingGroup(12);</span>
<matrix group of size 479001600 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">BasicSpinRepresentationOfSymmetricGroup( 10, 3, -1 );</span>
[ < immutable compressed matrix 16x16 over GF(9) >,
< immutable compressed matrix 16x16 over GF(9) >,
< immutable compressed matrix 16x16 over GF(9) >,
< immutable compressed matrix 16x16 over GF(9) >,
< immutable compressed matrix 16x16 over GF(9) >,
< immutable compressed matrix 16x16 over GF(9) >,
< immutable compressed matrix 16x16 over GF(9) >,
< immutable compressed matrix 16x16 over GF(9) >,
< immutable compressed matrix 16x16 over GF(9) > ]
</pre></div>
<p><a id="X7DDA6BC1824F78FD" name="X7DDA6BC1824F78FD"></a></p>
<h5>39.24-9 BasicSpinRepresentationOfSymmetricGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasicSpinRepresentationOfSymmetricGroup</code>( <var class="Arg">n</var>, <var class="Arg">p</var>, <var class="Arg">sign</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Constructs the image of the Coxeter generators in the basic spin (projective) representation of the symmetric group of degree <var class="Arg">n</var> over a field of characteristic <span class="SimpleMath">\(\textit{p} \geq 0\)</span>. There are two such representations and <var class="Arg">sign</var> controls which is returned: +1 gives a group where the preimage of an adjacent transposition <span class="SimpleMath">\((i,i+1)\)</span> has order 4, -1 gives a group where the preimage of an adjacent transposition <span class="SimpleMath">\((i,i+1)\)</span> has order 2. If no <var class="Arg">sign</var> is specified, +1 is used by default. If no <var class="Arg">p</var> is specified, 3 is used by default. (Note that the convention of which cover is labelled as +1 is inconsistent in the literature.)</p>
<p><a id="X844CFFDE80F6AD15" name="X844CFFDE80F6AD15"></a></p>
<h5>39.24-10 SchurCoverOfSymmetricGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SchurCoverOfSymmetricGroup</code>( <var class="Arg">n</var>, <var class="Arg">p</var>, <var class="Arg">sign</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Constructs a Schur cover of <code class="code">SymmetricGroup(<var class="Arg">n</var>)</code> as a faithful, irreducible matrix group in characteristic <var class="Arg">p</var> (<span class="SimpleMath">\(\textit{p} \neq 2\)</span>). For <span class="SimpleMath">\(\textit{n} \geq 4\)</span>, there are two such covers, and <var class="Arg">sign</var> determines which is returned: +1 gives a group where the preimage of an adjacent transposition <span class="SimpleMath">\((i,i+1)\)</span> has order 4, -1 gives a group where the preimage of an adjacent transposition <span class="SimpleMath">\((i,i+1)\)</span> has order 2. If no <var class="Arg">sign</var> is specified, +1 is used by default. If no <var class="Arg">p</var> is specified, 3 is used by default. (Note that the convention of which cover is labelled as +1 is inconsistent in the literature.) For <span class="SimpleMath">\(\textit{n} \leq 3\)</span>, the symmetric group is its own Schur cover and <var class="Arg">sign</var> is ignored. For <span class="SimpleMath">\(\textit{p} = 2\)</span>, there is no faithful, irreducible representation of the Schur cover unless <span class="SimpleMath">\(\textit{n} = 1\)</span> or <span class="SimpleMath">\(\textit{n} = 3\)</span>, so <code class="keyw">fail</code> is returned if <span class="SimpleMath">\(\textit{p} = 2\)</span>. For <span class="SimpleMath">\(\textit{p} = 3\)</span>, <span class="SimpleMath">\(\textit{n} = 3\)</span>, the representation is indecomposable, but reducible. The field of the matrix group is generally <code class="code">GF(<var class="Arg">p</var>^2)</code> if <span class="SimpleMath">\(\textit{p} > 0\)</span>, and an abelian number field if <span class="SimpleMath">\(\textit{p} = 0\)</span>.</p>
<p><a id="X7E0F4896795E34FC" name="X7E0F4896795E34FC"></a></p>
<h5>39.24-11 DoubleCoverOfAlternatingGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DoubleCoverOfAlternatingGroup</code>( <var class="Arg">n</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Constructs a double cover of <code class="code">AlternatingGroup(<var class="Arg">n</var>)</code> as a faithful, completely reducible matrix group in characteristic <var class="Arg">p</var> (<span class="SimpleMath">\(p \neq 2\)</span>) for <span class="SimpleMath">\(n \geq 4\)</span>. For <span class="SimpleMath">\(n \leq 3\)</span>, the alternating group is its own Schur cover, and <code class="keyw">fail</code> is returned. For <span class="SimpleMath">\(p = 2\)</span>, there is no faithful, completely reducible representation of the double cover, so <code class="keyw">fail</code> is returned. The field of the matrix group is generally <code class="code">GF(p^2)</code> if <span class="SimpleMath">\(p>0\)</span>, and an abelian number field if <span class="SimpleMath">\(p=0\)</span>. If <var class="Arg">p</var> is omitted, the default is 3.</p>
<p><a id="X865722987E0E19B6" name="X865722987E0E19B6"></a></p>
<h4>39.25 <span class="Heading">Tests for the Availability of Methods</span></h4>
<p>The following filters and operations indicate capabilities of <strong class="pkg">GAP</strong>. They can be used in the method selection or algorithms to check whether it is feasible to compute certain operations for a given group. In general, they return <code class="keyw">true</code> if good algorithms for the given arguments are available in <strong class="pkg">GAP</strong>. An answer <code class="keyw">false</code> indicates that no method for this group may exist, or that the existing methods might run into problems.</p>
<p>Typical examples when this might happen is with finitely presented groups, for which many of the methods cannot be guaranteed to succeed in all situations.</p>
<p>The willingness of <strong class="pkg">GAP</strong> to perform certain operations may change, depending on which further information is known about the arguments. Therefore the filters used are not implemented as properties but as "other filters" (see <a href="chap13_mj.html#X871597447BB998A1"><span class="RefLink">13.7</span></a> and <a href="chap13_mj.html#X7997705185C7E720"><span class="RefLink">13.8</span></a>).</p>
<p><a id="X798F13EA810FB215" name="X798F13EA810FB215"></a></p>
<h5>39.25-1 CanEasilyTestMembership</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanEasilyTestMembership</code>( <var class="Arg">G</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>This filter indicates whether <strong class="pkg">GAP</strong> can test membership of elements in the group <var class="Arg">G</var> (via the operation <code class="func">\in</code> (<a href="chap30_mj.html#X84B7FA8C7C94400F"><span class="RefLink">30.6-1</span></a>)) in reasonable time. It is used by the method selection to decide whether an algorithm that relies on membership tests may be used.</p>
<p><a id="X7C2A89607BDFD920" name="X7C2A89607BDFD920"></a></p>
<h5>39.25-2 CanEasilyComputeWithIndependentGensAbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanEasilyComputeWithIndependentGensAbelianGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>This filter indicates whether <strong class="pkg">GAP</strong> can in reasonable time compute independent abelian generators of the group <var class="Arg">G</var> (via <code class="func">IndependentGeneratorsOfAbelianGroup</code> (<a href="chap39_mj.html#X7D1574457B152333"><span class="RefLink">39.22-5</span></a>)) and then can decompose arbitrary group elements with respect to these generators using <code class="func">IndependentGeneratorExponents</code> (<a href="chap39_mj.html#X86F835DA8264A0CE"><span class="RefLink">39.22-6</span></a>). It is used by the method selection to decide whether an algorithm that relies on these two operations may be used.</p>
<p><a id="X83245C82835D496C" name="X83245C82835D496C"></a></p>
<h5>39.25-3 CanComputeSize</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanComputeSize</code>( <var class="Arg">dom</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>This filter indicates that we know that the size of the domain <var class="Arg">dom</var> (which might be <code class="func">infinity</code> (<a href="chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>)) can be computed reasonably easily. It doesn't imply as quick a computation as <code class="code">HasSize</code> would but its absence does not imply that the size cannot be computed.</p>
<p><a id="X8268965487364912" name="X8268965487364912"></a></p>
<h5>39.25-4 CanComputeSizeAnySubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanComputeSizeAnySubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>This filter indicates whether <strong class="pkg">GAP</strong> can easily compute the size of any subgroup of the group <var class="Arg">G</var>. (This is for example advantageous if one can test that a stabilizer index equals the length of the orbit computed so far to stop early.)</p>
<p><a id="X82DDE00D82A32083" name="X82DDE00D82A32083"></a></p>
<h5>39.25-5 CanComputeIndex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanComputeIndex</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This function indicates whether the index <span class="SimpleMath">\([\textit{G}:\textit{H}]\)</span> (which might be <code class="func">infinity</code> (<a href="chap18_mj.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>)) can be computed. It assumes that <span class="SimpleMath">\(\textit{H} \leq \textit{G}\)</span> (see <code class="func">CanComputeIsSubset</code> (<a href="chap39_mj.html#X7BE7C36B84C23511"><span class="RefLink">39.25-6</span></a>)).</p>
<p><a id="X7BE7C36B84C23511" name="X7BE7C36B84C23511"></a></p>
<h5>39.25-6 CanComputeIsSubset</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanComputeIsSubset</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This filter indicates that <strong class="pkg">GAP</strong> can test (via <code class="func">IsSubset</code> (<a href="chap30_mj.html#X79CA175481F8105F"><span class="RefLink">30.5-1</span></a>)) whether <var class="Arg">B</var> is a subset of <var class="Arg">A</var>.</p>
<p><a id="X87D62C2C7C375E2D" name="X87D62C2C7C375E2D"></a></p>
<h5>39.25-7 KnowsHowToDecompose</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KnowsHowToDecompose</code>( <var class="Arg">G</var>[, <var class="Arg">gens</var>] )</td><td class="tdright">( property )</td></tr></table></div>
<p>Tests whether the group <var class="Arg">G</var> can decompose elements in the generators <var class="Arg">gens</var>. If <var class="Arg">gens</var> is not given it tests, whether it can decompose in the generators given in the <code class="func">GeneratorsOfGroup</code> (<a href="chap39_mj.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>) value of <var class="Arg">G</var>.</p>
<p>This property can be used for example to check whether a group homomorphism by images (see <code class="func">GroupHomomorphismByImages</code> (<a href="chap40_mj.html#X7F348F497C813BE0"><span class="RefLink">40.1-1</span></a>)) can be reasonably defined from this group.</p>
<p><a id="X83A9997586694DC0" name="X83A9997586694DC0"></a></p>
<h4>39.26 <span class="Heading">Specific functions for Normalizer calculation</span></h4>
<p><a id="X84ABCA997D294B36" name="X84ABCA997D294B36"></a></p>
<h5>39.26-1 NormalizerViaRadical</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizerViaRadical</code>( <var class="Arg">G</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function implements a particular approach, following the SolvableRadical paradigm, for calculating the normalizer of a subgroup <var class="Arg">S</var> in <var class="Arg">G</var>. It is at the moment provided only as a separate function, and not as method for the operation <code class="code">Normalizer</code>, as it can often be slower than other built-in routines. In certain hard cases (non-solvable groups with nontrivial radical), however its performance is substantially superior. The function thus is provided as a non-automated tool for advanced users.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=TransitiveGroup(30,2030);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=SylowSubgroup(g,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(NormalizerViaRadical(g,s));</span>
28800
</pre></div>
<p>Note that this example only demonstrates usage, but that in this case in fact the ordinary <code class="code">Normalizer</code> routine performs faster.</p>
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