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# XMod

## Crossed modules and cat1-groups in GAP

Version 2.41

05/06/2015

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

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.

### 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;

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

- the actor crossed square of a crossed module.

Initial, experimental versions of the following are also included in this version:

- crossed squares and their morphisms;

- crossed modules of groupoids.

XMod was originally implemented in 1997 using the GAP3 language. when the first 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.

Version 2.19 was released on 9th June 2012 for the first release of GAP 4.5.

The current version is 2.41, released 5th June 2015 for GAP 4.7.

Bug reports, suggestions and comments are, of course, welcome. Please contact the second author at c.d.wensley@bangor.ac.uk.

### Copyright

© 1997-2015 Murat Alp and Chris Wensley. 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 of Frank Lübeck and Max Neunhöffer.

The first 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 Size
2.1-4 SubXMod
2.2 Pre-crossed modules
2.2-1 PreXModByBoundaryAndAction
2.2-2 PeifferSubgroup
2.2-3 IsPermXMod
2.3 Cat1-groups and pre-cat1-groups
2.3-1 Source
2.3-2 Cat1
2.3-3 Cat1OfXMod
2.4 Selection of a small cat1-group
2.4-1 Cat1Select
2.4-2 AllCat1sBasic
3 2d-mappings
3.1 Morphisms of 2d-groups
3.1-1 Source
3.2 Morphisms of pre-crossed modules
3.2-1 IsXModMorphism
3.2-2 IsInjective
3.2-3 XModMorphism
3.3 Morphisms of pre-cat1-groups
3.3-1 IsCat1Morphism
3.4 Operations on morphisms
3.4-1 CompositionMorphism
3.4-2 Kernel
4 Derivations and Sections
4.1 Whitehead Multiplication
4.1-1 IsDerivation
4.1-2 DerivationByImages
4.1-3 SectionByImages
4.2 Whitehead Groups and Monoids
4.2-1 RegularDerivations
4.2-2 CompositeDerivation
4.2-3 WhiteheadGroupTable
5 Actors of 2d-groups
5.1 Actor of a crossed module
5.1-1 WhiteheadXMod
5.1-2 XModCentre
6 Induced Constructions
6.1 Induced crossed modules
6.1-1 InducedXMod
6.1-2 AllInducedXMods
7 Crossed squares and their morphisms
7.1 Constructions for crossed squares
7.1-1 XSq
7.1-2 IsXSq
7.1-3 Up2dGroup
7.2 Morphisms of crossed squares
7.2-1 Source
7.2-2 IsXSqMorphism
8 Crossed modules of groupoids
8.1 Constructions for crossed modules of groupoids
8.1-1 PreXModWithObjectsObj
9 Utility functions
9.1 Printing Lists
9.1-1 PrintListOneItemPerLine
9.2 Inclusion and Restriction Mappings
9.2-1 InclusionMappingGroups
9.2-2 InnerAutomorphismByNormalSubgroup
9.3 Abelian Modules
9.3-1 AbelianModuleObject
9.4 Distinct and Common Representatives
9.4-1 DistinctRepresentatives
10 Development history
10.1 Changes from version to version
10.1-1 Version 1 for GAP 3
10.1-2 Version 2
10.1-3 Version 2.001 for GAP 4
10.1-4 Induced crossed modules
10.1-5 Versions 2.002 -- 2.006
10.1-6 Versions 2.007 -- 2.010
10.2 Versions for GAP 4.5, 4.6 and 4.7
10.2-1 AllCat1s
10.2-2 Version 2.41
10.3 What needs doing next?
References
Index
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