chap0.txt

  
  
                                      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:[email protected]
      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:[email protected]
      Address:  Nigde Üniversitesi,
                Fen Edebiyat Fakültesi, Matematik Bölümü
                Nigde, Turkey.
  
  
  Alper Odabaş
      Email:    mailto:[email protected]
      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:[email protected].
  
  
  -------------------------------------------------------
  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?