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# XMod

## Crossed modules and cat1-groups in GAP

Version 2.64

30/11/2017

Chris Wensley
Email: c.d.wensley@bangor.ac.uk
Homepage: http://pages.bangor.ac.uk/~mas023/
Address:
School of Computer Science, Bangor University,
Dean Street, Bangor, Gwynedd, LL57 1UT, U.K.

Murat Alp
Email: muratalp@nigde.edu.tr
Address:
Nigde Üniversitesi,
Fen Edebiyat Fakültesi, Matematik Bölümü
Nigde, Turkey.

Alper Odabaş
Email: aodabas@ogu.edu.tr
Homepage: http://fef.ogu.edu.tr/matbil/aodabas/
Address:
Department of Mathematics and Computer Science,
Osmangazi University, Eskişehir, Turkey

Enver Önder Uslu
Address:
Department of Mathematics and Computer Science,
Osmangazi University, Eskişehir, Turkey

### 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 1997 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].

The current version is 2.64, released 30th November 2017 for GAP 4.8.

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

© 1997-2017 Chris Wensley et al. XMod 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 package [LN17] of Frank Lübeck and Max Neunhöffer.

The procedure used to mount new releases on GitHub uses the packages GitHubPagesForGAP [Hor17] and ReleaseTools of Max Horn.

The second author wishes to acknowledge support from Dumlupinar University and the Turkish government.

### 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 Derivations and Sections
5.1 Whitehead Multiplication
5.1-1 IsDerivation
5.1-2 DerivationByImages
5.1-3 SectionByImages
5.2 Whitehead Groups and Monoids
5.2-1 RegularDerivations
5.2-2 CompositeDerivation
5.2-3 WhiteheadGroupTable
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
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 Utility functions
10.1 Inclusion and Restriction Mappings
10.1-1 InclusionMappingGroups
10.1-2 InnerAutomorphismsByNormalSubgroup
10.2 Abelian Modules
10.2-1 AbelianModuleObject
11 Development history
11.1 Changes from version to version
11.1-1 Version 1 for GAP 3
11.1-2 Version 2
11.1-3 Version 2.001 for GAP 4
11.1-4 Induced crossed modules
11.1-5 Versions 2.002 -- 2.006
11.1-6 Versions 2.007 -- 2.010
11.2 Versions for GAP [4.5 .. 4.8]
11.2-1 AllCat1s
11.2-2 Versions 2.43 - 2.56
11.2-3 Version 2.61
11.2-4 Latest Version
11.3 What needs doing next?
References
Index
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