XMod Crossed modules and cat1-groups in GAP Version 2.64 30/11/2017 Chris Wensley Murat Alp Alper Odabaş Enver Önder Uslu Chris Wensley Email: mailto: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: mailto:muratalp@nigde.edu.tr Address: Nigde Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü Nigde, Turkey. Alper Odabaş Email: mailto:aodabas@ogu.edu.tr Homepage: http://fef.ogu.edu.tr/matbil/aodabas/ 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 mailto: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 (http://www.fsf.org/licenses/gpl.html) 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 (XMod) 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?