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Polycyclic

Computation with polycyclic groups

2.17

28 August 2025

Bettina Eick
Email: 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: mhorn@rptu.de
Homepage: https://www.quendi.de/math
Address:
Fachbereich Mathematik
RPTU Kaiserslautern-Landau
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 the GNU General Public License 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 and contributions to this package and its documentation provided by GAP users and developers.

Contents

2 Introduction to polycyclic presentations

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

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

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