<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >script type="text/javascript" "https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML " >script >title >GAP (polycyclic) - Contents</title >meta http-equiv="content-type" content="text/html; charset=UTF-8" />meta name="generator" content="GAPDoc2HTML" />link rel="stylesheet" type="text/css" href="manual.css" />script src="manual.js" type="text/javascript" ></script >script type="text/javascript" >overwriteStyle();</script >head >body class="chap0" onload="jscontent()" >div class="chlinktop" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chap5_mj.html" >5</a> <a href="chap6_mj.html" >6</a> <a href="chap7_mj.html" >7</a> <a href="chap8_mj.html" >8</a> <a href="chap9_mj.html" >9</a> <a href="chapA_mj.html" >A</a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" >Ind</a> </div >div class="chlinkprevnexttop" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap1_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap0.html" >[MathJax off]</a></p>"X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5" ></a></p>div class="pcenter" >h1 >Polycyclic</h1 >div >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 />br />Homepage: <span class="URL" ><a href="http://www.mathematik.tu-darmstadt.de/~nickel/ " >http://www.mathematik.tu-darmstadt.de/~nickel/</a></span >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 />"X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>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>"X82A988D47DFAFCFA" name="X82A988D47DFAFCFA" ></a></p>ackage and its documentation provided by <strong class="pkg" >GAP</strong > users and developers.</p>"X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>div class="contents" >"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 >div ></div >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 >div ></div >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 >div ></div >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 >div ></div >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 >div ></div >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 >div ></div >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 >div ></div >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 >div ></div >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 >div ></div >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 >div ></div >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 >div ></div >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 >div ></div >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 >div ></div >div >div class="ContChap" ><a href="chapA_mj.html#X874ECE907CAF380D" >A <span class="Heading" >Obsolete Functions and Name Changes</span ></a>div >div class="ContChap" ><a href="chapBib_mj.html" ><span class="Heading" >References</span ></a></div >div 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