Polycyclic   Computation with polycyclic groups  2.16 25 July 2020 Bettina Eick Werner Nickel Max Horn Bettina Eick Email: mailto:beick@tu-bs.de Homepage: http://www.iaa.tu-bs.de/beick Address: Institut Analysis und Algebra TU Braunschweig Universitätsplatz 2 D-38106 Braunschweig Germany Werner Nickel Homepage: http://www.mathematik.tu-darmstadt.de/~nickel/ Max Horn Email: mailto:horn@mathematik.uni-kl.de Homepage: https://www.quendi.de/math Address: Fachbereich Mathematik TU Kaiserslautern Gottlieb-Daimler-Straße 48 67663 Kaiserslautern Germany ------------------------------------------------------- Copyright © 2003-2018 by Bettina Eick, Max Horn and Werner Nickel The Polycyclic package is free software;you can redistribute it and/or modify it under the terms of theGNU General Public License (http://www.fsf.org/licenses/gpl.html)as published by the Free Software Foundation; either version 2 of the License,or (at your option) any later version. ------------------------------------------------------- Acknowledgements We appreciate very much all past and future comments, suggestions andcontributions to this package and its documentation provided by GAPusers and developers. ------------------------------------------------------- Contents (polycyclic) 1 Preface 2 Introduction to polycyclic presentations 3 Collectors 3.1 Constructing a Collector 3.1-1 FromTheLeftCollector 3.1-2 SetRelativeOrder 3.1-3 SetPower 3.1-4 SetConjugate 3.1-5 SetCommutator 3.1-6 UpdatePolycyclicCollector 3.1-7 IsConfluent 3.2 Accessing Parts of a Collector 3.2-1 RelativeOrders 3.2-2 GetPower 3.2-3 GetConjugate 3.2-4 NumberOfGenerators 3.2-5 ObjByExponents 3.2-6 ExponentsByObj 3.3 Special Features 3.3-1 IsWeightedCollector 3.3-2 AddHallPolynomials 3.3-3 String 3.3-4 FTLCollectorPrintTo 3.3-5 FTLCollectorAppendTo 3.3-6 UseLibraryCollector 3.3-7 USE_LIBRARY_COLLECTOR 3.3-8 DEBUG_COMBINATORIAL_COLLECTOR 3.3-9 USE_COMBINATORIAL_COLLECTOR 4 Pcp-groups - polycyclically presented groups 4.1 Pcp-elements -- elements of a pc-presented group 4.1-1 PcpElementByExponentsNC 4.1-2 PcpElementByGenExpListNC 4.1-3 IsPcpElement 4.1-4 IsPcpElementCollection 4.1-5 IsPcpElementRep 4.1-6 IsPcpGroup 4.2 Methods for pcp-elements 4.2-1 Collector 4.2-2 Exponents 4.2-3 GenExpList 4.2-4 NameTag 4.2-5 Depth 4.2-6 LeadingExponent 4.2-7 RelativeOrder 4.2-8 RelativeIndex 4.2-9 FactorOrder 4.2-10 NormingExponent 4.2-11 NormedPcpElement 4.3 Pcp-groups - groups of pcp-elements 4.3-1 PcpGroupByCollector 4.3-2 Group 4.3-3 Subgroup 5 Basic methods and functions for pcp-groups 5.1 Elementary methods for pcp-groups 5.1-1 \= 5.1-2 Size 5.1-3 Random 5.1-4 Index 5.1-5 \in 5.1-6 Elements 5.1-7 ClosureGroup 5.1-8 NormalClosure 5.1-9 HirschLength 5.1-10 CommutatorSubgroup 5.1-11 PRump 5.1-12 SmallGeneratingSet 5.2 Elementary properties of pcp-groups 5.2-1 IsSubgroup 5.2-2 IsNormal 5.2-3 IsNilpotentGroup 5.2-4 IsAbelian 5.2-5 IsElementaryAbelian 5.2-6 IsFreeAbelian 5.3 Subgroups of pcp-groups 5.3-1 Igs 5.3-2 Ngs 5.3-3 Cgs 5.3-4 SubgroupByIgs 5.3-5 AddToIgs 5.4 Polycyclic presentation sequences for subfactors 5.4-1 Pcp 5.4-2 GeneratorsOfPcp 5.4-3 \[\] 5.4-4 Length 5.4-5 RelativeOrdersOfPcp 5.4-6 DenominatorOfPcp 5.4-7 NumeratorOfPcp 5.4-8 GroupOfPcp 5.4-9 OneOfPcp 5.4-10 ExponentsByPcp 5.4-11 PcpGroupByPcp 5.5 Factor groups of pcp-groups 5.5-1 NaturalHomomorphismByNormalSubgroup 5.5-2 \/ 5.6 Homomorphisms for pcp-groups 5.6-1 GroupHomomorphismByImages 5.6-2 Kernel 5.6-3 Image 5.6-4 PreImage 5.6-5 PreImagesRepresentative 5.6-6 IsInjective 5.7 Changing the defining pc-presentation 5.7-1 RefinedPcpGroup 5.7-2 PcpGroupBySeries 5.8 Printing a pc-presentation 5.8-1 PrintPcpPresentation 5.9 Converting to and from a presentation 5.9-1 IsomorphismPcpGroup 5.9-2 IsomorphismPcpGroupFromFpGroupWithPcPres 5.9-3 IsomorphismPcGroup 5.9-4 IsomorphismFpGroup 6 Libraries and examples of pcp-groups 6.1 Libraries of various types of polycyclic groups 6.1-1 AbelianPcpGroup 6.1-2 DihedralPcpGroup 6.1-3 UnitriangularPcpGroup 6.1-4 SubgroupUnitriangularPcpGroup 6.1-5 InfiniteMetacyclicPcpGroup 6.1-6 HeisenbergPcpGroup 6.1-7 MaximalOrderByUnitsPcpGroup 6.1-8 BurdeGrunewaldPcpGroup 6.2 Some assorted example groups 6.2-1 ExampleOfMetabelianPcpGroup 6.2-2 ExamplesOfSomePcpGroups 7 Higher level methods for pcp-groups 7.1 Subgroup series in pcp-groups 7.1-1 PcpSeries 7.1-2 EfaSeries 7.1-3 SemiSimpleEfaSeries 7.1-4 DerivedSeriesOfGroup 7.1-5 RefinedDerivedSeries 7.1-6 RefinedDerivedSeriesDown 7.1-7 LowerCentralSeriesOfGroup 7.1-8 UpperCentralSeriesOfGroup 7.1-9 TorsionByPolyEFSeries 7.1-10 PcpsBySeries 7.1-11 PcpsOfEfaSeries 7.2 Orbit stabilizer methods for pcp-groups 7.2-1 PcpOrbitStabilizer 7.2-2 StabilizerIntegralAction 7.2-3 NormalizerIntegralAction 7.3 Centralizers, Normalizers and Intersections 7.3-1 Centralizer 7.3-2 Centralizer 7.3-3 Intersection 7.4 Finite subgroups 7.4-1 TorsionSubgroup 7.4-2 NormalTorsionSubgroup 7.4-3 IsTorsionFree 7.4-4 FiniteSubgroupClasses 7.4-5 FiniteSubgroupClassesBySeries 7.5 Subgroups of finite index and maximal subgroups 7.5-1 MaximalSubgroupClassesByIndex 7.5-2 LowIndexSubgroupClasses 7.5-3 LowIndexNormalSubgroups 7.5-4 NilpotentByAbelianNormalSubgroup 7.6 Further attributes for pcp-groups based on the Fitting subgroup 7.6-1 FittingSubgroup 7.6-2 IsNilpotentByFinite 7.6-3 Centre 7.6-4 FCCentre 7.6-5 PolyZNormalSubgroup 7.6-6 NilpotentByAbelianByFiniteSeries 7.7 Functions for nilpotent groups 7.7-1 MinimalGeneratingSet 7.8 Random methods for pcp-groups 7.8-1 RandomCentralizerPcpGroup 7.9 Non-abelian tensor product and Schur extensions 7.9-1 SchurExtension 7.9-2 SchurExtensionEpimorphism 7.9-3 SchurCover 7.9-4 AbelianInvariantsMultiplier 7.9-5 NonAbelianExteriorSquareEpimorphism 7.9-6 NonAbelianExteriorSquare 7.9-7 NonAbelianTensorSquareEpimorphism 7.9-8 NonAbelianTensorSquare 7.9-9 NonAbelianExteriorSquarePlusEmbedding 7.9-10 NonAbelianTensorSquarePlusEpimorphism 7.9-11 NonAbelianTensorSquarePlus 7.9-12 WhiteheadQuadraticFunctor 7.10 Schur covers 7.10-1 SchurCovers 8 Cohomology for pcp-groups 8.1 Cohomology records 8.1-1 CRRecordByMats 8.1-2 CRRecordBySubgroup 8.2 Cohomology groups 8.2-1 OneCoboundariesCR 8.2-2 TwoCohomologyModCR 8.3 Extended 1-cohomology 8.3-1 OneCoboundariesEX 8.3-2 OneCocyclesEX 8.3-3 OneCohomologyEX 8.4 Extensions and Complements 8.4-1 ComplementCR 8.4-2 ComplementsCR 8.4-3 ComplementClassesCR 8.4-4 ComplementClassesEfaPcps 8.4-5 ComplementClasses 8.4-6 ExtensionCR 8.4-7 ExtensionsCR 8.4-8 ExtensionClassesCR 8.4-9 SplitExtensionPcpGroup 8.5 Constructing pcp groups as extensions 9 Matrix Representations 9.1 Unitriangular matrix groups 9.1-1 UnitriangularMatrixRepresentation 9.1-2 IsMatrixRepresentation 9.2 Upper unitriangular matrix groups 9.2-1 IsomorphismUpperUnitriMatGroupPcpGroup 9.2-2 SiftUpperUnitriMatGroup 9.2-3 RanksLevels 9.2-4 MakeNewLevel 9.2-5 SiftUpperUnitriMat 9.2-6 DecomposeUpperUnitriMat A Obsolete Functions and Name Changes