  
  
                                     [1X XMod [101X
  
  
                       [1X Crossed Modules and Cat1-Groups [101X
  
  
                                      2.82
  
  
                                23 October 2020
  
  
                                 Chris Wensley
  
                                   Murat Alp
  
                                  Alper Odabas
  
                                Enver Onder Uslu
  
  
  
  Chris Wensley
      Email:    [7Xmailto:c.d.wensley@bangor.ac.uk[107X
      Homepage: [7Xhttp://pages.bangor.ac.uk/~mas023/[107X
      Address:  [33X[0;14YDr. C.D. Wensley[133X
                [33X[0;14YSchool of Computer Science and Electronic Engineering[133X
                [33X[0;14YBangor University[133X
                [33X[0;14YDean Street[133X
                [33X[0;14YBangor[133X
                [33X[0;14YGwynedd LL57 1UT[133X
                [33X[0;14YUK[133X
  
  
  Murat Alp
      Email:    [7Xmailto:muratalp@nigde.edu.tr[107X
      Address:  [33X[0;14YProf. Dr. M. Alp[133X
                [33X[0;14YÖmer Halisdemir University[133X
                [33X[0;14YArt and Science Faculty[133X
                [33X[0;14YMathematics Department[133X
                [33X[0;14YNigde[133X
                [33X[0;14YTurkey[133X
  
  
  Alper Odabas
      Email:    [7Xmailto:aodabas@ogu.edu.tr[107X
      Address:  [33X[0;14YDr. A. Odabas[133X
                [33X[0;14YOsmangazi University[133X
                [33X[0;14YArts and Sciences Faculty[133X
                [33X[0;14YDepartment of Mathematics and Computer Science[133X
                [33X[0;14YEskisehir[133X
                [33X[0;14YTurkey[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 1996 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;0YBug reports, suggestions and comments are, of course, welcome. Please submit
  an issue at [7Xhttps://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© 1996-2020, Chris Wensley et al.[133X
  
  [33X[0;0YThe  [5XXMod[105X package 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.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThis  documentation  was prepared using the [5XGAPDoc[105X [LN17] and [5XAutoDoc[105X [GH17]
  packages.[133X
  
  [33X[0;0YThe   procedure   used   to   produce   new   releases   uses   the  package
  [5XGitHubPagesForGAP[105X [Hor17] and the package [5XReleaseTools[105X.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (XMod)[101X
  
  1 [33X[0;0YIntroduction[133X
  2 [33X[0;0Y2d-groups : crossed modules and cat[22X^1[122X-groups[133X
    2.1 [33X[0;0YConstructions for crossed modules[133X
      2.1-1 XMod
      2.1-2 XModByNormalSubgroup
      2.1-3 XModByTrivialAction
      2.1-4 XModByAutomorphismGroup
      2.1-5 XModByCentralExtension
      2.1-6 XModByPullback
      2.1-7 XModByAbelianModule
      2.1-8 DirectProduct
      2.1-9 Source
      2.1-10 ImageElmXModAction
      2.1-11 Size
    2.2 [33X[0;0YProperties of crossed modules[133X
      2.2-1 IsXMod
      2.2-2 SubXMod
      2.2-3 KernelCokernelXMod
    2.3 [33X[0;0YPre-crossed modules[133X
      2.3-1 PreXModByBoundaryAndAction
      2.3-2 PeifferSubgroup
    2.4 [33X[0;0YCat[22X^1[122X-groups and pre-cat[22X^1[122X-groups[133X
      2.4-1 Cat1Group
      2.4-2 Source
      2.4-3 DiagonalCat1Group
      2.4-4 TransposeCat1Group
      2.4-5 Cat1GroupByPeifferQuotient
      2.4-6 SubCat1Group
      2.4-7 DirectProduct
    2.5 [33X[0;0YProperties of cat[22X^1[122X-groups and pre-cat[22X^1[122X-groups[133X
      2.5-1 IsCat1Group
      2.5-2 IsPreCat1GroupWithIdentityEmbedding
      2.5-3 Cat1GroupOfXMod
    2.6 [33X[0;0YEnumerating cat[22X^1[122X-groups with a given source[133X
      2.6-1 AllCat1GroupsWithImageIterator
      2.6-2 AllCat1GroupsMatrix
      2.6-3 AllCat1GroupsIterator
      2.6-4 CatnGroupNumbers
    2.7 [33X[0;0YSelection of a small cat[22X^1[122X-group[133X
      2.7-1 Cat1Select
    2.8 [33X[0;0YMore functions for crossed modules and cat[22X^1[122X-groups[133X
      2.8-1 IdGroup
      2.8-2 IsSubXMod
    2.9 [33X[0;0YThe group groupoid associated to a cat[22X^1[122X-group[133X
      2.9-1 GroupGroupoid
      2.9-2 GroupGroupoidElement
  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
      3.2-5 MorphismOfPullback
    3.3 [33X[0;0YMorphisms of pre-cat1-groups[133X
      3.3-1 IsCat1GroupMorphism
      3.3-2 Cat1GroupMorphismOfXModMorphism
      3.3-3 IsomorphismPermObject
      3.3-4 SmallerDegreePermutationRepresentation2DimensionalGroup
    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 IsomorphismXMods
      4.1-12 AllXMods
    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;0YWhitehead group of a crossed module[133X
    5.1 [33X[0;0YDerivations and Sections[133X
      5.1-1 DerivationByImages
      5.1-2 PrincipalDerivation
      5.1-3 SectionByHomomorphism
      5.1-4 IdentityDerivation
      5.1-5 WhiteheadProduct
    5.2 [33X[0;0YWhitehead Groups and Monoids[133X
      5.2-1 AllDerivations
      5.2-2 RegularDerivations
      5.2-3 PrincipalDerivations
      5.2-4 AllSections
  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
    7.3 [33X[0;0YInduced cat[22X^1[122X-groups[133X
      7.3-1 InducedCat1Group
  8 [33X[0;0YCrossed 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 CrossedSquareByXMods
      8.2-2 CrossedSquareByNormalSubgroups
      8.2-3 CrossedSquareByNormalSubXMod
      8.2-4 ActorCrossedSquare
      8.2-5 CrossedSquareByAutomorphismGroup
      8.2-6 CrossedSquareByPullback
      8.2-7 CrossedSquareByXModSplitting
      8.2-8 CrossedSquare
      8.2-9 Transpose3DimensionalGroup
      8.2-10 CentralQuotient
      8.2-11 IsCrossedSquare
      8.2-12 Up2DimensionalGroup
      8.2-13 IsSymmetric3DimensionalGroup
      8.2-14 CrossedPairing
    8.3 [33X[0;0YMorphisms of crossed squares[133X
      8.3-1 CrossedSquareMorphism
      8.3-2 Source
      8.3-3 IsCrossedSquareMorphism
      8.3-4 InclusionMorphismHigherDimensionalDomains
    8.4 [33X[0;0YDefinitions and constructions for cat[22X^2[122X-groups and their morphisms[133X
      8.4-1 Cat2Group
      8.4-2 DirectProduct
      8.4-3 DisplayLeadMaps
      8.4-4 Transpose3DimensionalGroup
      8.4-5 Cat2GroupMorphism
      8.4-6 Cat2GroupOfCrossedSquare
      8.4-7 Subdiagonal2DimensionalGroup
    8.5 [33X[0;0YEnumerating cat[22X^2[122X-groups with a given source[133X
      8.5-1 AllCat2GroupsWithImagesIterator
      8.5-2 AllCat2GroupsWithFixedUp
      8.5-3 AllCat2GroupsMatrix
      8.5-4 AllCat2GroupsIterator
  9 [33X[0;0YCrossed cubes and Cat[22X^3[122X-groups[133X
    9.1 [33X[0;0YFunctions for (pre-)cat[22X^3[122X-groups[133X
      9.1-1 Cat3Group
      9.1-2 Front3DimensionalGroup
    9.2 [33X[0;0YEnumerating cat[22X^3[122X-groups with a given source[133X
      9.2-1 AllCat3GroupTriples
    9.3 [33X[0;0YDefinition and constructions for cat[22X^n[122X-groups and their morphisms[133X
      9.3-1 PreCatnGroup
  10 [33X[0;0YCrossed modules of groupoids[133X
    10.1 [33X[0;0YConstructions for crossed modules of groupoids[133X
      10.1-1 SinglePiecePreXModWithObjects
      10.1-2 IsXModWithObjects
      10.1-3 IsPermPreXModWithObjects
      10.1-4 Root2dGroup
  11 [33X[0;0YApplications[133X
    11.1 [33X[0;0YFree Loop Spaces[133X
      11.1-1 LoopClasses
  12 [33X[0;0YInteraction with HAP[133X
    12.1 [33X[0;0YCalling HAP functions[133X
      12.1-1 SmallCat1Group
      12.1-2 CatOneGroupToXMod
      12.1-3 IdCat1Group
  13 [33X[0;0YUtility functions[133X
    13.1 [33X[0;0YMappings[133X
      13.1-1 InclusionMappingGroups
      13.1-2 InnerAutomorphismsByNormalSubgroup
      13.1-3 IsGroupOfAutomorphisms
    13.2 [33X[0;0YAbelian Modules[133X
      13.2-1 AbelianModuleObject
  14 [33X[0;0YDevelopment history[133X
    14.1 [33X[0;0YChanges from version to version[133X
      14.1-1 [33X[0;0YVersion 1 for [5XGAP[105X 3[133X
      14.1-2 [33X[0;0YVersion 2[133X
      14.1-3 [33X[0;0YVersion 2.001 for [5XGAP[105X 4[133X
      14.1-4 [33X[0;0YInduced crossed modules[133X
      14.1-5 [33X[0;0YVersions 2.002 -- 2.006[133X
      14.1-6 [33X[0;0YVersions 2.007 -- 2.010[133X
    14.2 [33X[0;0YVersions for [5XGAP[105X [4.5 .. 4.8][133X
      14.2-1 [33X[0;0YAllCat1s[133X
      14.2-2 [33X[0;0YVersions 2.43 - 2.56[133X
      14.2-3 [33X[0;0YVersion 2.61[133X
      14.2-4 [33X[0;0YVersion 2.63 - 2.74[133X
    14.3 [33X[0;0YWhat needs doing next?[133X
  
  
  [32X
