  
  
                                      [1X[5XXMod[105X[101X
  
  
                     [1XCrossed modules and cat1-groups in [5XGAP[105X[101X
  
  
                                  Version 2.64
  
  
                                   30/11/2017
  
  
                                 Chris Wensley
  
                                   Murat Alp
  
                                  Alper Odabaş
  
                                Enver Önder Uslu
  
  
  
  Chris Wensley
      Email:    [7Xmailto:c.d.wensley@bangor.ac.uk[107X
      Homepage: [7Xhttp://pages.bangor.ac.uk/~mas023/[107X
      Address:  [33X[0;14YSchool of Computer Science, Bangor University,[133X
                [33X[0;14YDean Street, Bangor, Gwynedd, LL57 1UT, U.K.[133X
  
  
  Murat Alp
      Email:    [7Xmailto:muratalp@nigde.edu.tr[107X
      Address:  [33X[0;14YNigde Üniversitesi,[133X
                [33X[0;14YFen Edebiyat Fakültesi, Matematik Bölümü[133X
                [33X[0;14YNigde, Turkey.[133X
  
  
  Alper Odabaş
      Email:    [7Xmailto:aodabas@ogu.edu.tr[107X
      Homepage: [7Xhttp://fef.ogu.edu.tr/matbil/aodabas/[107X
      Address:  [33X[0;14YDepartment of Mathematics and Computer Science,[133X
                [33X[0;14YOsmangazi University, Eskişehir, Turkey[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe [5XXMod[105X package provides functions for computation with[133X
  
  [30X    [33X[0;6Yfinite  crossed  modules  of  groups and cat1-groups, and morphisms of
        these structures;[133X
  
  [30X    [33X[0;6Yfinite   pre-crossed   modules,  pre-cat1-groups,  and  their  Peiffer
        quotients;[133X
  
  [30X    [33X[0;6Yisoclinism classes of groups and crossed modules;[133X
  
  [30X    [33X[0;6Yderivations of crossed modules and sections of cat1-groups;[133X
  
  [30X    [33X[0;6Ycrossed  squares  and  their  morphisms,  including  the actor crossed
        square of a crossed module;[133X
  
  [30X    [33X[0;6Ycrossed modules of finite groupoids (experimental version).[133X
  
  [33X[0;0Y[5XXMod[105X  was  originally  implemented in 1997 using the [5XGAP[105X3 language, when the
  second author was studying for a Ph.D. [Alp97] in Bangor.[133X
  
  [33X[0;0YIn  April  2002  the  first  and  third  parts  were  converted to [5XGAP[105X4, 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 [5XGAP[105X 4.4.[133X
  
  [33X[0;0YIn  October  2015  functions  for  computing  isoclinism  classes of crossed
  modules,  written  by  Alper  Odabaş  and  Enver Uslu, were added. These are
  contained in Chapter [14X4[114X, and are described in detail in the paper [IOU16].[133X
  
  [33X[0;0YThe current version is 2.64, released 30th November 2017 for [5XGAP[105X 4.8.[133X
  
  [33X[0;0YBug reports, suggestions and comments are, of course, welcome. Please submit
  an  issue at [7Xhttp://github.com/gap-packages/xmod/issues/[107X or send an email to
  the first author at [7Xmailto:c.d.wensley@bangor.ac.uk[107X.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 1997-2017 Chris Wensley et al. [5XXMod[105X is free software; you can redistribute
  it  and/or  modify  it  under  the  terms  of the GNU General Public License
  ([7Xhttp://www.fsf.org/licenses/gpl.html[107X)  as  published  by  the Free Software
  Foundation;  either  version 2 of the License, or (at your option) any later
  version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThis  documentation  was  prepared  with  the [5XGAPDoc[105X package [LN17] of Frank
  Lübeck and Max Neunhöffer.[133X
  
  [33X[0;0YThe  procedure  used  to  mount  new  releases  on  GitHub uses the packages
  [5XGitHubPagesForGAP[105X [Hor17] and [5XReleaseTools[105X of Max Horn.[133X
  
  [33X[0;0YThe  second  author wishes to acknowledge support from Dumlupinar University
  and the Turkish government.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (XMod)[101X
  
  1 [33X[0;0YIntroduction[133X
  2 [33X[0;0Y2d-groups : crossed modules and cat1-groups[133X
    2.1 [33X[0;0YConstructions for crossed modules[133X
      2.1-1 XMod
      2.1-2 Source
      2.1-3 ImageElmXModAction
      2.1-4 Size
    2.2 [33X[0;0YProperties of crossed modules[133X
      2.2-1 IsXMod
      2.2-2 SubXMod
    2.3 [33X[0;0YPre-crossed modules[133X
      2.3-1 PreXModByBoundaryAndAction
      2.3-2 PeifferSubgroup
    2.4 [33X[0;0YCat1-groups and pre-cat1-groups[133X
      2.4-1 Cat1Group
      2.4-2 Source
      2.4-3 DiagonalCat1Group
    2.5 [33X[0;0YProperties of cat1-groups and pre-cat1-groups[133X
      2.5-1 IsCat1Group
      2.5-2 Cat1GroupOfXMod
    2.6 [33X[0;0YSelection of a small cat1-group[133X
      2.6-1 Cat1Select
      2.6-2 AllCat1DataGroupsBasic
    2.7 [33X[0;0YMore functions for crossed modules and cat1-groups[133X
      2.7-1 IdGroup
  3 [33X[0;0Y2d-mappings[133X
    3.1 [33X[0;0YMorphisms of 2-dimensional groups[133X
      3.1-1 Source
    3.2 [33X[0;0YMorphisms of pre-crossed modules[133X
      3.2-1 IsXModMorphism
      3.2-2 IsInjective
      3.2-3 XModMorphism
      3.2-4 IsomorphismPerm2DimensionalGroup
    3.3 [33X[0;0YMorphisms of pre-cat1-groups[133X
      3.3-1 IsCat1Morphism
      3.3-2 IsomorphismPermObject
    3.4 [33X[0;0YOperations on morphisms[133X
      3.4-1 CompositionMorphism
      3.4-2 Kernel
  4 [33X[0;0YIsoclinism of groups and crossed modules[133X
    4.1 [33X[0;0YMore operations for crossed modules[133X
      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 [33X[0;0YIsoclinism for groups[133X
      4.2-1 Isoclinism
      4.2-2 IsStemDomain
      4.2-3 IsoclinicRank
    4.3 [33X[0;0YIsoclinism for crossed modules[133X
      4.3-1 Isoclinism
      4.3-2 IsStemDomain
      4.3-3 IsoclinicRank
  5 [33X[0;0YDerivations and Sections[133X
    5.1 [33X[0;0YWhitehead Multiplication[133X
      5.1-1 IsDerivation
      5.1-2 DerivationByImages
      5.1-3 SectionByImages
    5.2 [33X[0;0YWhitehead Groups and Monoids[133X
      5.2-1 RegularDerivations
      5.2-2 CompositeDerivation
      5.2-3 WhiteheadGroupTable
  6 [33X[0;0YActors of 2d-groups[133X
    6.1 [33X[0;0YActor of a crossed module[133X
      6.1-1 AutomorphismPermGroup
      6.1-2 WhiteheadXMod
      6.1-3 XModCentre
  7 [33X[0;0YInduced constructions[133X
    7.1 [33X[0;0YCoproducts of crossed modules[133X
      7.1-1 CoproductXMod
    7.2 [33X[0;0YInduced crossed modules[133X
      7.2-1 InducedXMod
      7.2-2 AllInducedXMods
  8 [33X[0;0Y3d-groups and 3d-mappings : crossed squares and cat[22X^2[122X-groups[133X
    8.1 [33X[0;0YDefinition of a crossed square and a crossed [22Xn[122X-cube of groups[133X
    8.2 [33X[0;0YConstructions for crossed squares[133X
      8.2-1 CrossedSquare
      8.2-2 CentralQuotient
      8.2-3 IsCrossedSquare
      8.2-4 Up2DimensionalGroup
    8.3 [33X[0;0YMorphisms of crossed squares[133X
      8.3-1 Source
      8.3-2 IsCrossedSquareMorphism
    8.4 [33X[0;0YDefinitions and constructions for cat[22X^2[122X-groups and their morphisms[133X
      8.4-1 Cat2Group
      8.4-2 Cat2GroupOfCrossedSquare
    8.5 [33X[0;0YDefinition and constructions for cat[22X^n[122X-groups and their morphisms[133X
  9 [33X[0;0YCrossed modules of groupoids[133X
    9.1 [33X[0;0YConstructions for crossed modules of groupoids[133X
      9.1-1 SinglePiecePreXModWithObjects
      9.1-2 IsXModWithObjects
      9.1-3 IsPermPreXModWithObjects
      9.1-4 Root2dGroup
  10 [33X[0;0YUtility functions[133X
    10.1 [33X[0;0YInclusion and Restriction Mappings[133X
      10.1-1 InclusionMappingGroups
      10.1-2 InnerAutomorphismsByNormalSubgroup
    10.2 [33X[0;0YAbelian Modules[133X
      10.2-1 AbelianModuleObject
  11 [33X[0;0YDevelopment history[133X
    11.1 [33X[0;0YChanges from version to version[133X
      11.1-1 [33X[0;0YVersion 1 for [5XGAP[105X 3[133X
      11.1-2 [33X[0;0YVersion 2[133X
      11.1-3 [33X[0;0YVersion 2.001 for [5XGAP[105X 4[133X
      11.1-4 [33X[0;0YInduced crossed modules[133X
      11.1-5 [33X[0;0YVersions 2.002 -- 2.006[133X
      11.1-6 [33X[0;0YVersions 2.007 -- 2.010[133X
    11.2 [33X[0;0YVersions for [5XGAP[105X [4.5 .. 4.8][133X
      11.2-1 [33X[0;0YAllCat1s[133X
      11.2-2 [33X[0;0YVersions 2.43 - 2.56[133X
      11.2-3 [33X[0;0YVersion 2.61[133X
      11.2-4 [33X[0;0YLatest Version[133X
    11.3 [33X[0;0YWhat needs doing next?[133X
  
  
  [32X
