[1X Polycyclic [101X
[1X Computation with polycyclic groups [101X
2.16
25 July 2020
Bettina Eick
Werner Nickel
Max Horn
Bettina Eick
Email: [7Xmailto:beick@tu-bs.de[107X
Homepage: [7Xhttp://www.iaa.tu-bs.de/beick[107X
Address: [33X[0;14YInstitut Analysis und Algebra[133X
[33X[0;14YTU Braunschweig[133X
[33X[0;14YUniversitätsplatz 2[133X
[33X[0;14YD-38106 Braunschweig[133X
[33X[0;14YGermany[133X
Werner Nickel
Homepage: [7Xhttp://www.mathematik.tu-darmstadt.de/~nickel/[107X
Max Horn
Email: [7Xmailto:horn@mathematik.uni-kl.de[107X
Homepage: [7Xhttps://www.quendi.de/math[107X
Address: [33X[0;14YFachbereich Mathematik[133X
[33X[0;14YTU Kaiserslautern[133X
[33X[0;14YGottlieb-Daimler-Straße 48[133X
[33X[0;14Y67663 Kaiserslautern[133X
[33X[0;14YGermany[133X
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[1XCopyright[101X
[33X[0;0Y© 2003-2018 by Bettina Eick, Max Horn and Werner Nickel[133X
[33X[0;0YThe [5XPolycyclic[105X package is free software;you can redistribute it and/or
modify it under the terms of theGNU General Public License
([7Xhttp://www.fsf.org/licenses/gpl.html[107X)as published by the Free Software
Foundation; either version 2 of the License,or (at your option) any later
version.[133X
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[1XAcknowledgements[101X
[33X[0;0YWe appreciate very much all past and future comments, suggestions
andcontributions to this package and its documentation provided by [5XGAP[105Xusers
and developers.[133X
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[1XContents (polycyclic)[101X
1 [33X[0;0YPreface[133X
2 [33X[0;0YIntroduction to polycyclic presentations[133X
3 [33X[0;0YCollectors[133X
3.1 [33X[0;0YConstructing a Collector[133X
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 [33X[0;0YAccessing Parts of a Collector[133X
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 [33X[0;0YSpecial Features[133X
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 [33X[0;0YPcp-groups - polycyclically presented groups[133X
4.1 [33X[0;0YPcp-elements -- elements of a pc-presented group[133X
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 [33X[0;0YMethods for pcp-elements[133X
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 [33X[0;0YPcp-groups - groups of pcp-elements[133X
4.3-1 PcpGroupByCollector
4.3-2 Group
4.3-3 Subgroup
5 [33X[0;0YBasic methods and functions for pcp-groups[133X
5.1 [33X[0;0YElementary methods for pcp-groups[133X
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 [33X[0;0YElementary properties of pcp-groups[133X
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 [33X[0;0YSubgroups of pcp-groups[133X
5.3-1 Igs
5.3-2 Ngs
5.3-3 Cgs
5.3-4 SubgroupByIgs
5.3-5 AddToIgs
5.4 [33X[0;0YPolycyclic presentation sequences for subfactors[133X
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 [33X[0;0YFactor groups of pcp-groups[133X
5.5-1 NaturalHomomorphismByNormalSubgroup
5.5-2 \/
5.6 [33X[0;0YHomomorphisms for pcp-groups[133X
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 [33X[0;0YChanging the defining pc-presentation[133X
5.7-1 RefinedPcpGroup
5.7-2 PcpGroupBySeries
5.8 [33X[0;0YPrinting a pc-presentation[133X
5.8-1 PrintPcpPresentation
5.9 [33X[0;0YConverting to and from a presentation[133X
5.9-1 IsomorphismPcpGroup
5.9-2 IsomorphismPcpGroupFromFpGroupWithPcPres
5.9-3 IsomorphismPcGroup
5.9-4 IsomorphismFpGroup
6 [33X[0;0YLibraries and examples of pcp-groups[133X
6.1 [33X[0;0YLibraries of various types of polycyclic groups[133X
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 [33X[0;0YSome assorted example groups[133X
6.2-1 ExampleOfMetabelianPcpGroup
6.2-2 ExamplesOfSomePcpGroups
7 [33X[0;0YHigher level methods for pcp-groups[133X
7.1 [33X[0;0YSubgroup series in pcp-groups[133X
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 [33X[0;0YOrbit stabilizer methods for pcp-groups[133X
7.2-1 PcpOrbitStabilizer
7.2-2 StabilizerIntegralAction
7.2-3 NormalizerIntegralAction
7.3 [33X[0;0YCentralizers, Normalizers and Intersections[133X
7.3-1 Centralizer
7.3-2 Centralizer
7.3-3 Intersection
7.4 [33X[0;0YFinite subgroups[133X
7.4-1 TorsionSubgroup
7.4-2 NormalTorsionSubgroup
7.4-3 IsTorsionFree
7.4-4 FiniteSubgroupClasses
7.4-5 FiniteSubgroupClassesBySeries
7.5 [33X[0;0YSubgroups of finite index and maximal subgroups[133X
7.5-1 MaximalSubgroupClassesByIndex
7.5-2 LowIndexSubgroupClasses
7.5-3 LowIndexNormalSubgroups
7.5-4 NilpotentByAbelianNormalSubgroup
7.6 [33X[0;0YFurther attributes for pcp-groups based on the Fitting subgroup[133X
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 [33X[0;0YFunctions for nilpotent groups[133X
7.7-1 MinimalGeneratingSet
7.8 [33X[0;0YRandom methods for pcp-groups[133X
7.8-1 RandomCentralizerPcpGroup
7.9 [33X[0;0YNon-abelian tensor product and Schur extensions[133X
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 [33X[0;0YSchur covers[133X
7.10-1 SchurCovers
8 [33X[0;0YCohomology for pcp-groups[133X
8.1 [33X[0;0YCohomology records[133X
8.1-1 CRRecordByMats
8.1-2 CRRecordBySubgroup
8.2 [33X[0;0YCohomology groups[133X
8.2-1 OneCoboundariesCR
8.2-2 TwoCohomologyModCR
8.3 [33X[0;0YExtended 1-cohomology[133X
8.3-1 OneCoboundariesEX
8.3-2 OneCocyclesEX
8.3-3 OneCohomologyEX
8.4 [33X[0;0YExtensions and Complements[133X
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 [33X[0;0YConstructing pcp groups as extensions[133X
9 [33X[0;0YMatrix Representations[133X
9.1 [33X[0;0YUnitriangular matrix groups[133X
9.1-1 UnitriangularMatrixRepresentation
9.1-2 IsMatrixRepresentation
9.2 [33X[0;0YUpper unitriangular matrix groups[133X
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 [33X[0;0YObsolete Functions and Name Changes[133X
[32X
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