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lpres

Nilpotent Quotients of L-Presented Groups

1.1.1

12 July 2024

René Hartung

Contents

1 The lpres package

1.1 Introduction

2 An Introduction to L-presented groups

2.1 Definitions

2.2 Creating an L-presented group

  2.2-1 LPresentedGroup
  2.2-2 ExamplesOfLPresentations
  2.2-3 FreeEngelGroup
  2.2-4 FreeBurnsideGroup
  2.2-5 FreeNilpotentGroup
  2.2-6 GeneralizedFabrykowskiGuptaLpGroup
  2.2-7 LamplighterGroup
  2.2-8 EmbeddingOfIASubgroup

2.3 The underlying free group

  2.3-1 FreeGroupOfLpGroup
  2.3-2 FreeGeneratorsOfLpGroup
  2.3-3 GeneratorsOfGroup
  2.3-4 UnderlyingElement
  2.3-5 ElementOfLpGroup

2.4 Accessing an L-presentation

  2.4-1 FixedRelatorsOfLpGroup
  2.4-2 IteratedRelatorsOfLpGroup
  2.4-3 EndomorphismsOfLpGroup

2.5 Attributes and properties of L-presented groups

  2.5-1 UnderlyingAscendingLPresentation
  2.5-2 UnderlyingInvariantLPresentation
  2.5-3 IsAscendingLPresentation
  2.5-4 IsInvariantLPresentation
  2.5-5 EmbeddingOfAscendingSubgroup

2.6 Methods for L-presented groups

  2.6-1 EpimorphismFromFpGroup
  2.6-2 SplitExtensionByAutomorphismsLpGroup
  2.6-3 AsLpGroup
  2.6-4 IsomorphismLpGroup

3 Nilpotent Quotients of L-presented groups

3.1 New methods for L-presented groups

  3.1-1 NilpotentQuotient
  3.1-2 LargestNilpotentQuotient
  3.1-3 NqEpimorphismNilpotentQuotient
  3.1-4 AbelianInvariants

3.2 A brief description of the algorithm

3.3 Nilpotent Quotient Systems for invariant L-presentations

3.3-1 InitQuotientSystem
3.3-2 ExtendQuotientSystem

3.4 Attributes of L-presented groups related with the nilpotent quotient algorithm

3.4-1 NilpotentQuotientSystem
3.4-2 NilpotentQuotients

3.5 The Info-Class InfoLPRES

3.5-1 InfoLPRES
3.5-2 InfoLPRES_MAX_GENS

4 Subgroups of L-presented groups

4.1 Creating a subgroup of an L-presented group

4.1-1 Subgroup
4.1-2 SubgroupLpGroupByCosetTable

4.2 Computing the index of finite-index subgroups

  4.2-1 IndexInWholeGroup
  4.2-2 Index
  4.2-3 CosetTableInWholeGroup

4.3 Technical details

4.3-1 LPRES_TCSTART
4.3-2 LPRES_CosetEnumerator

5 Approximating the Schur multiplier

  5.1-1 GeneratingSetOfMultiplier
  5.1-2 FiniteRankSchurMultiplier
  5.1-3 EndomorphismsOfFRSchurMultiplier
  5.1-4 EpimorphismCoveringGroups
  5.1-5 EpimorphismFiniteRankSchurMultiplier
  5.1-6 ImageInFiniteRankSchurMultiplier

6 On a parallel nilpotent quotient algorithm

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