  
  
                                  [1X Polycyclic [101X
  
  
                      [1X Computation with polycyclic groups [101X
  
  
                                      2.18
  
  
                                  9 April 2026
  
  
                                  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:mhorn@rptu.de[107X
      Homepage: [7Xhttps://www.quendi.de/math[107X
      Address:  [33X[0;14YFachbereich Mathematik[133X
                [33X[0;14YRPTU Kaiserslautern-Landau[133X
                [33X[0;14YGottlieb-Daimler-Straße 48[133X
                [33X[0;14Y67663 Kaiserslautern[133X
                [33X[0;14YGermany[133X
  
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2003-2026 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   the   GNU  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
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe  appreciate  very  much  all  past  and  future comments, suggestions and
  contributions  to  this  package and its documentation provided by [5XGAP[105X users
  and developers.[133X
  
  
  -------------------------------------------------------
  
  
  [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
