XMod

Crossed modules and cat1-groups in GAP

Version 2.008

October 2006

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

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Copyright

(C) 1997-2006 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 the Turkish government (add details?).

Colophon

XMod was originally implemented in 1997 using the GAP3 language. when the first author was studying for a Ph.D. [M97] 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 2.008, released on October 25th 2006.

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
   9.2 What needs doing next?




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