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<h1>lpres</h1>


<h2>Nilpotent Quotients of L-Presented Groups</h2>

<p>
    1.1.1</p>

<p>
    12 July 2024
  </p>

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<p><b>
    René Hartung

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<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X86B8787287B59CA4">1 <span class="Heading">The <strong class="pkg">lpres</strong> package</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7DFB63A97E67C0A1">1.1 <span class="Heading">Introduction</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X7AEB47327D75B633">2 <span class="Heading">An Introduction to L-presented groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X84541F61810C741D">2.1 <span class="Heading">Definitions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X81065E797A486D0F">2.2 <span class="Heading">Creating an L-presented group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7BBBE4C082AE4D5A">2.2-1 LPresentedGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79A034B8851444C9">2.2-2 ExamplesOfLPresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7DA323A87E7B6A7C">2.2-3 FreeEngelGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X81C3537083E40A5C">2.2-4 FreeBurnsideGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8796306C7A7924D1">2.2-5 FreeNilpotentGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X81450ABA81F0FCE5">2.2-6 GeneralizedFabrykowskiGuptaLpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X83BF8C597E1DC266">2.2-7 LamplighterGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7DBA63A37853BE46">2.2-8 EmbeddingOfIASubgroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X80B65AF48662DE70">2.3 <span class="Heading">The underlying free group</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7F883CC57A3CCAC7">2.3-1 FreeGroupOfLpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X838079A587E8CF43">2.3-2 FreeGeneratorsOfLpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79C44528864044C5">2.3-3 GeneratorsOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X85C405D57F65048A">2.3-4 UnderlyingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8573CDF57CB216D7">2.3-5 ElementOfLpGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X847047F083826C00">2.4 <span class="Heading">Accessing an L-presentation</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7CD9BE57815552FF">2.4-1 FixedRelatorsOfLpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C468D1C81964268">2.4-2 IteratedRelatorsOfLpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X85D253888263A3F6">2.4-3 EndomorphismsOfLpGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X817DA8E686311B54">2.5 <span class="Heading">Attributes and properties of L-presented groups</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X85E77B29796AB730">2.5-1 UnderlyingAscendingLPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X86F017E085082624">2.5-2 UnderlyingInvariantLPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X84E7A9E07A5DFDCF">2.5-3 IsAscendingLPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87F0C52978D99BB5">2.5-4 IsInvariantLPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X783B99E381C5C8BF">2.5-5 EmbeddingOfAscendingSubgroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7B5C48EA7CD8A57E">2.6 <span class="Heading">Methods for L-presented groups</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C81CB1C7F0D7A90">2.6-1 EpimorphismFromFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7972B0D87EF36536">2.6-2 SplitExtensionByAutomorphismsLpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X84F112247DA4037C">2.6-3 AsLpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X856F237B7BAC3BC8">2.6-4 IsomorphismLpGroup</a></span>
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<div class="ContChap"><a href="chap3.html#X824CC9CA824D3F1E">3 <span class="Heading">Nilpotent Quotients of L-presented groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X791C3E5280F38329">3.1 <span class="Heading">New methods for L-presented groups</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8216791583DE512C">3.1-1 NilpotentQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79AC8BE285CBB392">3.1-2 LargestNilpotentQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8758F663782AE655">3.1-3 NqEpimorphismNilpotentQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X812827937F403300">3.1-4 AbelianInvariants</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7C529DA9802E603E">3.2 <span class="Heading">A brief description of the algorithm</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X864A3F6F796E99DF">3.3 <span class="Heading">Nilpotent Quotient Systems for invariant L-presentations</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E58D47A8729FA8E">3.3-1 InitQuotientSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7910D0698781E02A">3.3-2 ExtendQuotientSystem</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X87CA2F188762A2B5">3.4 <span class="Heading">Attributes of L-presented groups related with the nilpotent quotient algorithm</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CC4586B85C22457">3.4-1 NilpotentQuotientSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D54126783CB7118">3.4-2 NilpotentQuotients</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7BB56B4C7C1EFAB8">3.5 <span class="Heading">The Info-Class InfoLPRES</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85F6BC1F8573D710">3.5-1 InfoLPRES</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X80F8139B81D2294E">3.5-2 InfoLPRES_MAX_GENS</a></span>
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<div class="ContChap"><a href="chap4.html#X874D64AA789F224E">4 <span class="Heading">Subgroups of L-presented groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X86B9E4BD7F5D1610">4.1 <span class="Heading">Creating a subgroup of an L-presented group</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C82AA387A42DCA0">4.1-1 Subgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7FC6C908782DEA48">4.1-2 SubgroupLpGroupByCosetTable</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A4EB4E0819ACB91">4.2 <span class="Heading">Computing the index of finite-index subgroups</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8014135884DCC53E">4.2-1 IndexInWholeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83A0356F839C696F">4.2-2 Index</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X846EC8AB7803114D">4.2-3 CosetTableInWholeGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X87A9EC0A7DF04931">4.3 <span class="Heading">Technical details</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X823EECA37A8EC3FE">4.3-1 LPRES_TCSTART</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C46A9B57BA4CA84">4.3-2 LPRES_CosetEnumerator</a></span>
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<div class="ContChap"><a href="chap5.html#X7FBE94957D7ECCFC">5 <span class="Heading">Approximating the Schur multiplier</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8606FDCE878850EF">5.1 <span class="Heading">Methods</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X83A5F95E84D3B662">5.1-1 GeneratingSetOfMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87A3D6C07D99C79A">5.1-2 FiniteRankSchurMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X78084374873BDFE1">5.1-3 EndomorphismsOfFRSchurMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7CF92D9880A3687E">5.1-4 EpimorphismCoveringGroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86EAE6457CE03B7B">5.1-5 EpimorphismFiniteRankSchurMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87182BC081DCA91E">5.1-6 ImageInFiniteRankSchurMultiplier</a></span>
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<div class="ContChap"><a href="chap6.html#X7BC16B0082A2B827">6 <span class="Heading">On a parallel nilpotent quotient algorithm</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X86A9B6F87E619FFF">6.1 <span class="Heading">Usage</span></a>
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<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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