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# XMod

## Crossed modules and cat1-groups in GAP

Version 2.12

November 2008

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

Chris Wensley
Email: c.d.wensley@bangor.ac.uk
Homepage: http://www.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 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).

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 2.12, released on 24th November 2008.

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

### Copyright

© 1997-2008 Murat Alp and Chris Wensley

### Acknowledgements

This xmod package is released under the GNU General Public License (GPL). This file is part of xmod, though as documentation it is released under the GNU Free Documentation License (see http://www.gnu.org/licenses/licenses.html#FDL).

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.

xmod is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with xmod; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.

For more details, see http://www.fsf.org/licenses/gpl.html.

This documentation was prepared with the GAPDoc package of Frank L\"ubeck and Max Neunh\"offer.

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

### 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 -- 2.010
9.1-7 Version 2.12
9.2 What needs doing next?
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