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# XMod

## Crossed Modules and Cat1-Groups

2.82

23 October 2020

Chris Wensley
Email: c.d.wensley@bangor.ac.uk
Homepage: http://pages.bangor.ac.uk/~mas023/
Address:
Dr. C.D. Wensley
School of Computer Science and Electronic Engineering
Bangor University
Dean Street
Bangor
Gwynedd LL57 1UT
UK

Murat Alp
Email: muratalp@nigde.edu.tr
Address:
Prof. Dr. M. Alp
Ömer Halisdemir University
Art and Science Faculty
Mathematics Department
Nigde
Turkey

Alper Odabas
Email: aodabas@ogu.edu.tr
Address:
Dr. A. Odabas
Osmangazi University
Arts and Sciences Faculty
Department of Mathematics and Computer Science
Eskisehir
Turkey

Enver Onder Uslu

### Abstract

The XMod package provides functions for computation with

- finite crossed modules of groups and cat1-groups, and morphisms of these structures;

- finite pre-crossed modules, pre-cat1-groups, and their Peiffer quotients;

- isoclinism classes of groups and crossed modules;

- derivations of crossed modules and sections of cat1-groups;

- crossed squares and their morphisms, including the actor crossed square of a crossed module;

- crossed modules of finite groupoids (experimental version).

XMod was originally implemented in 1996 using the GAP3 language, when the second author was studying for a Ph.D. [Alp97] in Bangor.

In April 2002 the first and third parts were converted to GAP4, the pre-structures were added, and version 2.001 was released. The final two parts, covering derivations, sections and actors, were included in the January 2004 release 2.002 for GAP 4.4.

In October 2015 functions for computing isoclinism classes of crossed modules, written by Alper Odabaş and Enver Uslu, were added. These are contained in Chapter 4, and are described in detail in the paper [IOU16].

Bug reports, suggestions and comments are, of course, welcome. Please submit an issue at https://github.com/gap-packages/xmod/issues/ or send an email to the first author at c.d.wensley@bangor.ac.uk.

### Copyright

© 1996-2020, Chris Wensley et al.

The XMod 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

This documentation was prepared using the GAPDoc [LN17] and AutoDoc [GH17] packages.

The procedure used to produce new releases uses the package GitHubPagesForGAP [Hor17] and the package ReleaseTools.

### Contents

1 Introduction
2 2d-groups : crossed modules and cat^1-groups
2.1 Constructions for crossed modules
2.1-1 XMod
2.1-2 XModByNormalSubgroup
2.1-3 XModByTrivialAction
2.1-4 XModByAutomorphismGroup
2.1-5 XModByCentralExtension
2.1-6 XModByPullback
2.1-7 XModByAbelianModule
2.1-8 DirectProduct
2.1-9 Source
2.1-10 ImageElmXModAction
2.1-11 Size
2.2 Properties of crossed modules
2.2-1 IsXMod
2.2-2 SubXMod
2.2-3 KernelCokernelXMod
2.3 Pre-crossed modules
2.3-1 PreXModByBoundaryAndAction
2.3-2 PeifferSubgroup
2.4 Cat^1-groups and pre-cat^1-groups
2.4-1 Cat1Group
2.4-2 Source
2.4-3 DiagonalCat1Group
2.4-4 TransposeCat1Group
2.4-5 Cat1GroupByPeifferQuotient
2.4-6 SubCat1Group
2.4-7 DirectProduct
2.5 Properties of cat^1-groups and pre-cat^1-groups
2.5-1 IsCat1Group
2.5-2 IsPreCat1GroupWithIdentityEmbedding
2.5-3 Cat1GroupOfXMod
2.6 Enumerating cat^1-groups with a given source
2.6-1 AllCat1GroupsWithImageIterator
2.6-2 AllCat1GroupsMatrix
2.6-3 AllCat1GroupsIterator
2.6-4 CatnGroupNumbers
2.7 Selection of a small cat^1-group
2.7-1 Cat1Select
2.8 More functions for crossed modules and cat^1-groups
2.8-1 IdGroup
2.8-2 IsSubXMod
2.9 The group groupoid associated to a cat^1-group
2.9-1 GroupGroupoid
2.9-2 GroupGroupoidElement
3 2d-mappings
3.1 Morphisms of 2-dimensional groups
3.1-1 Source
3.2 Morphisms of pre-crossed modules
3.2-1 IsXModMorphism
3.2-2 IsInjective
3.2-3 XModMorphism
3.2-4 IsomorphismPerm2DimensionalGroup
3.2-5 MorphismOfPullback
3.3 Morphisms of pre-cat1-groups
3.3-1 IsCat1GroupMorphism
3.3-2 Cat1GroupMorphismOfXModMorphism
3.3-3 IsomorphismPermObject
3.3-4 SmallerDegreePermutationRepresentation2DimensionalGroup
3.4 Operations on morphisms
3.4-1 CompositionMorphism
3.4-2 Kernel
4 Isoclinism of groups and crossed modules
4.1 More operations for crossed modules
4.1-1 FactorPreXMod
4.1-2 IntersectionSubXMods
4.1-3 Displacement
4.1-4 CommutatorSubXMod
4.1-5 DerivedSubXMod
4.1-6 FixedPointSubgroupXMod
4.1-7 CentreXMod
4.1-8 CentralQuotient
4.1-9 IsAbelian2DimensionalGroup
4.1-10 LowerCentralSeriesOfXMod
4.1-11 IsomorphismXMods
4.1-12 AllXMods
4.2 Isoclinism for groups
4.2-1 Isoclinism
4.2-2 IsStemDomain
4.2-3 IsoclinicRank
4.3 Isoclinism for crossed modules
4.3-1 Isoclinism
4.3-2 IsStemDomain
4.3-3 IsoclinicRank
5 Whitehead group of a crossed module
5.1 Derivations and Sections
5.1-1 DerivationByImages
5.1-2 PrincipalDerivation
5.1-3 SectionByHomomorphism
5.1-4 IdentityDerivation
5.1-5 WhiteheadProduct
5.2 Whitehead Groups and Monoids
5.2-1 AllDerivations
5.2-2 RegularDerivations
5.2-3 PrincipalDerivations
5.2-4 AllSections
6 Actors of 2d-groups
6.1 Actor of a crossed module
6.1-1 AutomorphismPermGroup
6.1-2 WhiteheadXMod
6.1-3 XModCentre
7 Induced constructions
7.1 Coproducts of crossed modules
7.1-1 CoproductXMod
7.2 Induced crossed modules
7.2-1 InducedXMod
7.2-2 AllInducedXMods
7.3 Induced cat^1-groups
7.3-1 InducedCat1Group
8 Crossed squares and Cat^2-groups
8.1 Definition of a crossed square and a crossed n-cube of groups
8.2 Constructions for crossed squares
8.2-1 CrossedSquareByXMods
8.2-2 CrossedSquareByNormalSubgroups
8.2-3 CrossedSquareByNormalSubXMod
8.2-4 ActorCrossedSquare
8.2-5 CrossedSquareByAutomorphismGroup
8.2-6 CrossedSquareByPullback
8.2-7 CrossedSquareByXModSplitting
8.2-8 CrossedSquare
8.2-9 Transpose3DimensionalGroup
8.2-10 CentralQuotient
8.2-11 IsCrossedSquare
8.2-12 Up2DimensionalGroup
8.2-13 IsSymmetric3DimensionalGroup
8.2-14 CrossedPairing
8.3 Morphisms of crossed squares
8.3-1 CrossedSquareMorphism
8.3-2 Source
8.3-3 IsCrossedSquareMorphism
8.3-4 InclusionMorphismHigherDimensionalDomains
8.4 Definitions and constructions for cat^2-groups and their morphisms
8.4-1 Cat2Group
8.4-2 DirectProduct
8.4-3 DisplayLeadMaps
8.4-4 Transpose3DimensionalGroup
8.4-5 Cat2GroupMorphism
8.4-6 Cat2GroupOfCrossedSquare
8.4-7 Subdiagonal2DimensionalGroup
8.5 Enumerating cat^2-groups with a given source
8.5-1 AllCat2GroupsWithImagesIterator
8.5-2 AllCat2GroupsWithFixedUp
8.5-3 AllCat2GroupsMatrix
8.5-4 AllCat2GroupsIterator
9 Crossed cubes and Cat^3-groups
9.1 Functions for (pre-)cat^3-groups
9.1-1 Cat3Group
9.1-2 Front3DimensionalGroup
9.2 Enumerating cat^3-groups with a given source
9.2-1 AllCat3GroupTriples
9.3 Definition and constructions for cat^n-groups and their morphisms
9.3-1 PreCatnGroup
10 Crossed modules of groupoids
10.1 Constructions for crossed modules of groupoids
10.1-1 SinglePiecePreXModWithObjects
10.1-2 IsXModWithObjects
10.1-3 IsPermPreXModWithObjects
10.1-4 Root2dGroup
11 Applications
11.1 Free Loop Spaces
11.1-1 LoopClasses
12 Interaction with HAP
12.1 Calling HAP functions
12.1-1 SmallCat1Group
12.1-2 CatOneGroupToXMod
12.1-3 IdCat1Group
13 Utility functions
13.1 Mappings
13.1-1 InclusionMappingGroups
13.1-2 InnerAutomorphismsByNormalSubgroup
13.1-3 IsGroupOfAutomorphisms
13.2 Abelian Modules
13.2-1 AbelianModuleObject
14 Development history
14.1 Changes from version to version
14.1-1 Version 1 for GAP 3
14.1-2 Version 2
14.1-3 Version 2.001 for GAP 4
14.1-4 Induced crossed modules
14.1-5 Versions 2.002 -- 2.006
14.1-6 Versions 2.007 -- 2.010
14.2 Versions for GAP [4.5 .. 4.8]
14.2-1 AllCat1s
14.2-2 Versions 2.43 - 2.56
14.2-3 Version 2.61
14.2-4 Version 2.63 - 2.74
14.3 What needs doing next?
References
Index
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