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# XMod

## Crossed modules and cat1-groups in GAP

Version 2.010

October 2007

Murat Alp
e-mail: malp@dumlupinar.edu.tr
Address:
Dumlupinar Universitesi,
Fen-Edebiyat Fakultesi, Matematik Bolumu
Merkez Kampus, Kutahya, Turkey.

Chris Wensley
e-mail: c.d.wensley@bangor.ac.uk
WWW: http://www.informatics.bangor.ac.uk/~cwensley/
Address:
School of Informatics, University of Wales Bangor,
Dean Street, Bangor, Gwynedd, LL57 1UT, U.K.

### Abstract

The XMod package provides functions for computation with

- finite crossed modules 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; and

- crossed squares and their morphisms (experimental version).

### Copyright

© 1997-2007 by Murat Alp and Chris Wensley

We adopt the copyright regulations of GAP as detailed in the copyright notice in the GAP manual.

### Acknowledgements

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

### Colophon

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.

The current version is 2.010, released on October 8th 2007.

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

### Contents

1. Introduction
2. 2d-objects
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
3. 2d-mappings
3.1 Morphisms of 2d-objects
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 Cat1Morphism
3.4 Operations on morphisms
3.4-1 Order
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-objects
5.1 Actor of a crossed module
5.1-1 WhiteheadXMod
5.1-2 Centre
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 Up2dObject
7.2 Morphisms of crossed squares
7.2-1 Source
7.2-2 IsXSqMorphism
8. Utility functions
8.1 Inclusion and Restriction Mappings
8.1-1 InclusionMappingGroups
8.2 Endomorphism Classes and Automorphisms
8.2-1 EndomorphismClasses
8.2-2 InnerAutomorphismByNormalSubgroup
8.3 Abelian Modules
8.3-1 AbelianModuleObject
8.4 Distinct and Common Representatives
8.4-1 DistinctRepresentatives
9. Development history
9.1 Changes from version to version
9.1-1 Version 1 for GAP 3
9.1-2 Version 2
9.1-3 Version 2.001 for GAP 4
9.1-4 Induced crossed modules
9.1-5 Versions 2.002 -- 2.006
9.1-6 Versions 2.007 -- 2.008
9.1-7 Version 2.010
9.2 What needs doing next?
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