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<h1>Polycyclic</h1>
<h2>Computation with polycyclic groups</h2>
<p>
2.17</p>
<p>
28 August 2025
</p>
</div>
<p><b>
Bettina Eick
</b>
<br />Email: <span class="URL"><a href="mailto:beick@tu-bs.de">beick@tu-bs.de</a></span>
<br />Homepage: <span class="URL"><a href="http://www.iaa.tu-bs.de/beick">http://www.iaa.tu-bs.de/beick</a></span>
<br />Address: <br />Institut Analysis und Algebra<br /> TU Braunschweig<br /> Universitätsplatz 2<br /> D-38106 Braunschweig<br /> Germany<br />
</p><p><b>
Werner Nickel
</b>
<br />Homepage: <span class="URL"><a href="http://www.mathematik.tu-darmstadt.de/~nickel/">http://www.mathematik.tu-darmstadt.de/~nickel/</a></span>
</p><p><b>
Max Horn
</b>
<br />Email: <span class="URL"><a href="mailto:mhorn@rptu.de">mhorn@rptu.de</a></span>
<br />Homepage: <span class="URL"><a href="https://www.quendi.de/math">https://www.quendi.de/math</a></span>
<br />Address: <br />Fachbereich Mathematik<br /> RPTU Kaiserslautern-Landau<br /> Gottlieb-Daimler-Straße 48<br /> 67663 Kaiserslautern<br /> Germany<br />
</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2003-2018 by Bettina Eick, Max Horn and Werner Nickel</p>
<p>The <strong class="pkg">Polycyclic</strong> package is free software; you can redistribute it and/or modify it under the terms of the <span class="URL"><a href="http://www.fsf.org/licenses/gpl.html">GNU General Public License</a></span> as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>We appreciate very much all past and future comments, suggestions and contributions to this package and its documentation provided by <strong class="pkg">GAP</strong> users and developers.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>
<div class="ContChap"><a href="chap1_mj.html#X874E1D45845007FE">1 <span class="Heading">Preface</span></a>
</div>
<div class="ContChap"><a href="chap2_mj.html#X792561B378D95B23">2 <span class="Heading">Introduction to polycyclic presentations</span></a>
</div>
<div class="ContChap"><a href="chap3_mj.html#X792305CC81E8606A">3 <span class="Heading">Collectors</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X800FD91386C08CD8">3.1 <span class="Heading">Constructing a Collector</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X8382A4E78706DE65">3.1-1 FromTheLeftCollector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X79A308B28183493B">3.1-2 SetRelativeOrder</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7BC319BA8698420C">3.1-3 SetPower</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X86A08D887E049347">3.1-4 SetConjugate</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7B25997C7DF92B6D">3.1-5 SetCommutator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7E9903F57BC5CC24">3.1-6 UpdatePolycyclicCollector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X8006790B86328CE8">3.1-7 IsConfluent</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X818484817C3BAAE6">3.2 <span class="Heading">Accessing Parts of a Collector</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7DD0DF677AC1CF10">3.2-1 RelativeOrders</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X844C0A478735EF4B">3.2-2 GetPower</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X865160E07FA93E00">3.2-3 GetConjugate</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7D6A26A4871FF51A">3.2-4 NumberOfGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X873ECF388503E5DE">3.2-5 ObjByExponents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X85BCB97B8021EAD6">3.2-6 ExponentsByObj</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X79AEB3477800DC16">3.3 <span class="Heading">Special Features</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X82EE2ACD7B8C178B">3.3-1 IsWeightedCollector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7A1D7ED68334282C">3.3-2 AddHallPolynomials</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X81FB5BE27903EC32">3.3-3 String</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7ED466B6807D16FE">3.3-4 FTLCollectorPrintTo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X789D9EB37ECFA9D7">3.3-5 FTLCollectorAppendTo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X808A26FB873A354F">3.3-6 UseLibraryCollector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X844E195C7D55F8BD">3.3-7 USE_LIBRARY_COLLECTOR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7945C6B97BECCDA8">3.3-8 DEBUG_COMBINATORIAL_COLLECTOR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7BDFB55D7CB33543">3.3-9 USE_COMBINATORIAL_COLLECTOR</a></span>
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</div>
<div class="ContChap"><a href="chap4_mj.html#X7E2AF25881CF7307">4 <span class="Heading">Pcp-groups - polycyclically presented groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7882F0F57ABEB680">4.1 <span class="Heading">Pcp-elements -- elements of a pc-presented group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X786DB93F7862D903">4.1-1 PcpElementByExponentsNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7BBB358C7AA64135">4.1-2 PcpElementByGenExpListNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X86083E297D68733B">4.1-3 IsPcpElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8695069A7D5073B7">4.1-4 IsPcpElementCollection</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F2C83AD862910B9">4.1-5 IsPcpElementRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8470284A78A6C41B">4.1-6 IsPcpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X790471D07A953E12">4.2 <span class="Heading">Methods for pcp-elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7E2D258B7DCE8AC9">4.2-1 Collector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X85C672E78630C507">4.2-2 Exponents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8571F6FB7E74346C">4.2-3 GenExpList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X82252C5E7B011559">4.2-4 NameTag</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X840D32D9837E99F5">4.2-5 Depth</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X874F1EC178721833">4.2-6 LeadingExponent</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8008AB61823A76B7">4.2-7 RelativeOrder</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X875D04288577015B">4.2-8 RelativeIndex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X87E070747955F2C1">4.2-9 FactorOrder</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X79A247797F0A8583">4.2-10 NormingExponent</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X798BB22B80833441">4.2-11 NormedPcpElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7A4EF7C68151905A">4.3 <span class="Heading">Pcp-groups - groups of pcp-elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7C8FBCAB7F63FACB">4.3-1 PcpGroupByCollector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D7B075385435151">4.3-2 Group</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7C82AA387A42DCA0">4.3-3 Subgroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5_mj.html#X7B9B85AE7C9B13EE">5 <span class="Heading">Basic methods and functions for pcp-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X821360107E355B88">5.1 <span class="Heading">Elementary methods for pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X806A4814806A4814"><code>5.1-1 \=</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X858ADA3B7A684421">5.1-2 Size</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79730D657AB219DB">5.1-3 Random</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X83A0356F839C696F">5.1-4 Index</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X87BDB89B7AAFE8AD"><code>5.1-5 \in</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79B130FC7906FB4C">5.1-6 Elements</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7D13FC1F8576FFD8">5.1-7 ClosureGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7BDEA0A98720D1BB">5.1-8 NormalClosure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X839B42AE7A1DD544">5.1-9 HirschLength</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7A9A3D5578CE33A0">5.1-10 CommutatorSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X796DA805853FAC90">5.1-11 PRump</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X814DBABC878D5232">5.1-12 SmallGeneratingSet</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X80E88168866D54F3">5.2 <span class="Heading">Elementary properties of pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7839D8927E778334">5.2-1 IsSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X838186F9836F678C">5.2-2 IsNormal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X87D062608719F2CD">5.2-3 IsNilpotentGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7C12AA7479A6C103">5.2-4 IsAbelian</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X813C952F80E775D4">5.2-5 IsElementaryAbelian</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X84FFC668832F9ED6">5.2-6 IsFreeAbelian</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X85A7E26C7E14AFBA">5.3 <span class="Heading">Subgroups of pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X815F756286701BE0">5.3-1 Igs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7F4D95C47F9652BA">5.3-2 Ngs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X8077293A787D4571">5.3-3 Cgs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X83B92A2679EAB1EB">5.3-4 SubgroupByIgs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X78107DE78728B26B">5.3-5 AddToIgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X803D62BC86EF07D0">5.4 <span class="Heading">Polycyclic presentation sequences for subfactors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7DD931697DD93169">5.4-1 Pcp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X821FF77086E38B3A">5.4-2 GeneratorsOfPcp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X8297BBCD79642BE6"><code>5.4-3 <span>\</span>[<span>\</span>]</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X780769238600AFD1">5.4-4 Length</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7ABCA7F2790E1673">5.4-5 RelativeOrdersOfPcp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7D16C299825887AA">5.4-6 DenominatorOfPcp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X803AED1A84FCBEE8">5.4-7 NumeratorOfPcp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X80BCCF0B81344933">5.4-8 GroupOfPcp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X87F0BA5F7BA0F4B4">5.4-9 OneOfPcp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7A8C8BBC81581E09">5.4-10 ExponentsByPcp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X87D75F7F86FEF203">5.4-11 PcpGroupByPcp</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X845D29B478CA7656">5.5 <span class="Heading">Factor groups of pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X80FC390C7F38A13F">5.5-1 NaturalHomomorphismByNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7F51DF007F51DF00"><code>5.5-2 \/</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X82E643F178E765EA">5.6 <span class="Heading">Homomorphisms for pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7F348F497C813BE0">5.6-1 GroupHomomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7DCD99628504B810">5.6-2 Kernel</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X847322667E6166C8">5.6-3 Image</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X836FAEAC78B55BF4">5.6-4 PreImage</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7AE24A1586B7DE79">5.6-5 PreImagesRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7F065FD7822C0A12">5.6-6 IsInjective</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7C873F807D4F3A3C">5.7 <span class="Heading">Changing the defining pc-presentation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X80E9B60E853B2E05">5.7-1 RefinedPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7F88F5548329E279">5.7-2 PcpGroupBySeries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X85E681027AF19B1E">5.8 <span class="Heading">Printing a pc-presentation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79D247127FD57FC8">5.8-1 PrintPcpPresentation</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X826ACBBB7A977206">5.9 <span class="Heading">Converting to and from a presentation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X8771540F7A235763">5.9-1 IsomorphismPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7F5EBF1C831B4BA9">5.9-2 IsomorphismPcpGroupFromFpGroupWithPcPres</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X873CEB137BA1CD6E">5.9-3 IsomorphismPcGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7F28268F850F454E">5.9-4 IsomorphismFpGroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6_mj.html#X78CEF1F27ED8D7BB">6 <span class="Heading">Libraries and examples of pcp-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X84A48FAB83934263">6.1 <span class="Heading">Libraries of various types of polycyclic groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7AEDE1BA82014B86">6.1-1 AbelianPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7ACF57737D0F12DB">6.1-2 DihedralPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X864CEDAB7911CC79">6.1-3 UnitriangularPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X812E35B17AADBCD5">6.1-4 SubgroupUnitriangularPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7A80F7F27FDA6810">6.1-5 InfiniteMetacyclicPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X81BEC875827D1CC2">6.1-6 HeisenbergPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X87F9B9C9786430D7">6.1-7 MaximalOrderByUnitsPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X852283A77A2C93DD">6.1-8 BurdeGrunewaldPcpGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X806FBA4A7CB8FB71">6.2 <span class="Heading">Some assorted example groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X86293081865CDFC3">6.2-1 ExampleOfMetabelianPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X83A74A6E7E232FD6">6.2-2 ExamplesOfSomePcpGroups</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap7_mj.html#X85BB6FE078679DAF">7 <span class="Heading">Higher level methods for pcp-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X8266A0A2821D98A1">7.1 <span class="Heading">Subgroup series in pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8037DAD77A19D9B2">7.1-1 PcpSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X86C633357ACD342C">7.1-2 EfaSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X80ED4F8380DC477E">7.1-3 SemiSimpleEfaSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7A879948834BD889">7.1-4 DerivedSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X866D4C5C79F26611">7.1-5 RefinedDerivedSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X86F7DE927DE3B5CD">7.1-6 RefinedDerivedSeriesDown</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X879D55A67DB42676">7.1-7 LowerCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8428592E8773CD7B">7.1-8 UpperCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X83CA5DE785AE3F2C">7.1-9 TorsionByPolyEFSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7E39431286969377">7.1-10 PcpsBySeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X79789A1C82139854">7.1-11 PcpsOfEfaSeries</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7CE2DA437FD2B383">7.2 <span class="Heading">Orbit stabilizer methods for pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X83E17DB483B33AB5">7.2-1 PcpOrbitStabilizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X80694BA480F69A0E">7.2-2 StabilizerIntegralAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X875BE4077B32A411">7.2-3 NormalizerIntegralAction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X80E3B42E792532B3">7.3 <span class="Heading">Centralizers, Normalizers and Intersections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X808EE8AD7EE3ECE1">7.3-1 Centralizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X849B5C527BAFAAA4">7.3-2 Centralizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X851069107CACF98E">7.3-3 Intersection</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7CF015E87A2B2388">7.4 <span class="Heading">Finite subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8036FA507A170DC4">7.4-1 TorsionSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8082CD337972DC63">7.4-2 NormalTorsionSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X86D92DA17DCE22DD">7.4-3 IsTorsionFree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X819058217B4F3DC0">7.4-4 FiniteSubgroupClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7E7C32EA81A297B6">7.4-5 FiniteSubgroupClassesBySeries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7D9F737F80F6E396">7.5 <span class="Heading">Subgroups of finite index and maximal subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X87D62D497A8715FB">7.5-1 MaximalSubgroupClassesByIndex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7800133F81BC7674">7.5-2 LowIndexSubgroupClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7F7067C77F2DC32C">7.5-3 LowIndexNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X85A5BC447D83175F">7.5-4 NilpotentByAbelianNormalSubgroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X785E0E877AB1D549">7.6 <span class="Heading">Further attributes for pcp-groups based on the Fitting subgroup</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X780552B57C30DD8F">7.6-1 FittingSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X86BD63DC844731DF">7.6-2 IsNilpotentByFinite</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X847ABE6F781C7FE8">7.6-3 Centre</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X861C36368435EB09">7.6-4 FCCentre</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7E75E2BC806746AC">7.6-5 PolyZNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X86800BF783E30D4A">7.6-6 NilpotentByAbelianByFiniteSeries</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X878DBDC77CCA4F7E">7.7 <span class="Heading">Functions for nilpotent groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X81D15723804771E2">7.7-1 MinimalGeneratingSet</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X8640F9D47A1F7434">7.8 <span class="Heading">Random methods for pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X80AEE73E7D639699">7.8-1 RandomCentralizerPcpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X824142B784453DB9">7.9 <span class="Heading">Non-abelian tensor product and Schur extensions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X79EF28D9845878C9">7.9-1 SchurExtension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X84B60EC978A9A05E">7.9-2 SchurExtensionEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7DD1E37987612042">7.9-3 SchurCover</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X792BC39D7CEB1D27">7.9-4 AbelianInvariantsMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X822ED5978647C93B">7.9-5 NonAbelianExteriorSquareEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8739CD4686301A0E">7.9-6 NonAbelianExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X86553D7B7DABF38F">7.9-7 NonAbelianTensorSquareEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7C0DF7C97F78C666">7.9-8 NonAbelianTensorSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7AE75EC1860FFE7A">7.9-9 NonAbelianExteriorSquarePlusEmbedding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7D96C84E87925B0F">7.9-10 NonAbelianTensorSquarePlusEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8746533787C4E8BC">7.9-11 NonAbelianTensorSquarePlus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X78F9184078B2761A">7.9-12 WhiteheadQuadraticFunctor</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7D3023697BA5CE5A">7.10 <span class="Heading">Schur covers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7D90B44E7B96AFF1">7.10-1 SchurCovers</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap8_mj.html#X796AB9787E2A752C">8 <span class="Heading">Cohomology for pcp-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X875758FA7C6F5CE1">8.1 <span class="Heading">Cohomology records</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7C97442C7B78806C">8.1-1 CRRecordByMats</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8646DFA1804D2A11">8.1-2 CRRecordBySubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X874759D582393441">8.2 <span class="Heading">Cohomology groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X85EF170387D39D4A">8.2-1 OneCoboundariesCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X79B48D697A8A84C8">8.2-2 TwoCohomologyModCR</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X79610E9178BD0C54">8.3 <span class="Heading">Extended 1-cohomology</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7E87E3EA81C84621">8.3-1 OneCoboundariesEX</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8111D2087C16CC0C">8.3-2 OneCocyclesEX</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X84718DDE792FB212">8.3-3 OneCohomologyEX</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X853E51787A24AE00">8.4 <span class="Heading">Extensions and Complements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7DA9162085058006">8.4-1 ComplementCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7F8984D386A813D6">8.4-2 ComplementsCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7FAB3EB0803197FA">8.4-3 ComplementClassesCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8759DC59799DD508">8.4-4 ComplementClassesEfaPcps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7B0EC76D81A056AB">8.4-5 ComplementClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X85F3B55C78CF840B">8.4-6 ExtensionCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X81DC85907E0948FD">8.4-7 ExtensionsCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7AE16E3687E14B24">8.4-8 ExtensionClassesCR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7986997B78AD3292">8.4-9 SplitExtensionPcpGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X823771527DBD857D">8.5 <span class="Heading">Constructing pcp groups as extensions</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap9_mj.html#X858D1BB07A8FBF87">9 <span class="Heading">Matrix Representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7D0ED06C7E6A457D">9.1 <span class="Heading">Unitriangular matrix groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7E6F320F865E309C">9.1-1 UnitriangularMatrixRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7F5E7F5F7DDB2E2C">9.1-2 IsMatrixRepresentation</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X79A8A51B84E4BF8C">9.2 <span class="Heading">Upper unitriangular matrix groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X8434972E7DDB68C1">9.2-1 IsomorphismUpperUnitriMatGroupPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X843C9D427FFA2487">9.2-2 SiftUpperUnitriMatGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7CF8B8F981931846">9.2-3 RanksLevels</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X81F3760186734EA7">9.2-4 MakeNewLevel</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X851A216C85B74574">9.2-5 SiftUpperUnitriMat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X86D711217C639C2C">9.2-6 DecomposeUpperUnitriMat</a></span>
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</div>
<div class="ContChap"><a href="chapA_mj.html#X874ECE907CAF380D">A <span class="Heading">Obsolete Functions and Name Changes</span></a>
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<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
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