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# XMod

## Crossed Modules and Cat1-Groups

2.72

16 September 2018

Christopher D. Wensley
Email: c.d.wensley@bangor.ac.uk
Homepage: http://pages.bangor.ac.uk/~mas023/
Address:
Dr. C.D. Wensley
School of Computer Science
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 http://github.com/gap-packages/xmod/issues/ or send an email to the first author at c.d.wensley@bangor.ac.uk.

### Copyright

© 1996-2018, 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 with 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 cat1-groups
2.1 Constructions for crossed modules
2.1-1 XMod
2.1-2 Source
2.1-3 ImageElmXModAction
2.1-4 Size
2.2 Properties of crossed modules
2.2-1 IsXMod
2.2-2 SubXMod
2.3 Pre-crossed modules
2.3-1 PreXModByBoundaryAndAction
2.3-2 PeifferSubgroup
2.4 Cat1-groups and pre-cat1-groups
2.4-1 Cat1Group
2.4-2 Source
2.4-3 DiagonalCat1Group
2.5 Properties of cat1-groups and pre-cat1-groups
2.5-1 IsCat1Group
2.5-2 Cat1GroupOfXMod
2.6 Selection of a small cat1-group
2.6-1 Cat1Select
2.6-2 AllCat1DataGroupsBasic
2.7 More functions for crossed modules and cat1-groups
2.7-1 IdGroup
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
3.3 Morphisms of pre-cat1-groups
3.3-1 IsCat1Morphism
3.3-2 IsomorphismPermObject
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 AllXMods
4.1-12 IsomorphismXMods
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 3d-groups and 3d-mappings : 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 CrossedSquare
8.2-2 CentralQuotient
8.2-3 IsCrossedSquare
8.2-4 Up2DimensionalGroup
8.3 Morphisms of crossed squares
8.3-1 Source
8.3-2 IsCrossedSquareMorphism
8.4 Definitions and constructions for cat^2-groups and their morphisms
8.4-1 Cat2Group
8.4-2 Cat2GroupOfCrossedSquare
8.5 Definition and constructions for cat^n-groups and their morphisms
9 Crossed modules of groupoids
9.1 Constructions for crossed modules of groupoids
9.1-1 SinglePiecePreXModWithObjects
9.1-2 IsXModWithObjects
9.1-3 IsPermPreXModWithObjects
9.1-4 Root2dGroup
10 Applications
10.1 Free Loop Spaces
10.1-1 LoopsXMod
11 Utility functions
11.1 Inclusion and Restriction Mappings
11.1-1 InclusionMappingGroups
11.1-2 InnerAutomorphismsByNormalSubgroup
11.2 Abelian Modules
11.2-1 AbelianModuleObject
12 Development history
12.1 Changes from version to version
12.1-1 Version 1 for GAP 3
12.1-2 Version 2
12.1-3 Version 2.001 for GAP 4
12.1-4 Induced crossed modules
12.1-5 Versions 2.002 -- 2.006
12.1-6 Versions 2.007 -- 2.010
12.2 Versions for GAP [4.5 .. 4.8]
12.2-1 AllCat1s
12.2-2 Versions 2.43 - 2.56
12.2-3 Version 2.61
12.2-4 Latest Version
12.3 What needs doing next?
References
Index
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