  
  
                                      [1X[5XXMod[0m[1X[0m
  
  
                     [1XCrossed modules and cat1-groups in [5XGAP[0m[1X[0m
  
  
                                  Version 2.12
  
  
                                 November 2008
  
  
                                   Murat Alp
  
                                 Chris Wensley
  
  
  
  Murat Alp
      Email:    [7Xmailto:malp@dumlupinar.edu.tr[0m
      Address:  Dumlupinar Universitesi,
                Fen-Edebiyat Fakultesi, Matematik Bolumu
                Merkez Kampus, Kutahya, Turkey.
  
  
  Chris Wensley
      Email:    [7Xmailto:c.d.wensley@bangor.ac.uk[0m
      Homepage: [7Xhttp://www.bangor.ac.uk/~mas023/[0m
      Address:  School of Computer Science, Bangor University,
                Dean Street, Bangor, Gwynedd, LL57 1UT, U.K.
  
  
  
  -------------------------------------------------------
  [1XAbstract[0m
  The XMod package provides functions for computation with
  
  --    finite  crossed  modules  and  cat1-groups,  and  morphisms  of  these
        structures;
  
  --    finite   pre-crossed   modules,  pre-cat1-groups,  and  their  Peiffer
        quotients;
  
  --    derivations of crossed modules and sections of cat1-groups;
  
  --    the actor crossed square of a crossed module; and
  
  --    crossed squares and their morphisms (experimental version).
  
  [5XXMod[0m  was  originally  implemented in 1997 using the [5XGAP[0m3 language. when the
  first author was studying for a Ph.D. [Alp97] in Bangor.
  
  In  April  2002  the  first  and  third  parts  were  converted to [5XGAP[0m4, 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 2.12, released on 24th November 2008.
  
  Bug  reports,  suggestions  and  comments  are,  of  course, welcome. Please
  contact the second author at [7Xmailto:c.d.wensley@bangor.ac.uk[0m.
  
  
  -------------------------------------------------------
  [1XCopyright[0m
  © 1997-2008 Murat Alp and Chris Wensley
  
  
  -------------------------------------------------------
  [1XAcknowledgements[0m
  This  [5Xxmod[0m  package  is released under the GNU General Public License (GPL).
  This  file is part of [5Xxmod[0m, though as documentation it is released under the
  GNU            Free           Documentation           License           (see
  [7Xhttp://www.gnu.org/licenses/licenses.html#FDL[0m).
  
  [5Xxmod[0m  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.
  
  [5Xxmod[0m  is  distributed  in  the  hope that it will be useful, but WITHOUT ANY
  WARRANTY;  without  even  the implied warranty of MERCHANTABILITY or FITNESS
  FOR  A  PARTICULAR  PURPOSE.  See  the  GNU  General Public License for more
  details.
  
  You should have received a copy of the GNU General Public License along with
  [5Xxmod[0m;  if not, write to the Free Software Foundation, Inc., 59 Temple Place,
  Suite 330, Boston, MA 02111-1307 USA.
  
  For more details, see [7Xhttp://www.fsf.org/licenses/gpl.html[0m.
  
  This  documentation  was  prepared with the [5XGAPDoc[0m package of Frank L\"ubeck
  and Max Neunh\"offer.
  
  The  first  author  wishes to acknowledge support from Dumlupinar University
  and the Turkish government.
  
  
  -------------------------------------------------------
  
  
  [1XContents (XMod)[0X
  
  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 [5XGAP[0m 3
      9.1-2 Version 2
      9.1-3 Version 2.001 for [5XGAP[0m 4
      9.1-4 Induced crossed modules
      9.1-5 Versions 2.002 -- 2.006
      9.1-6 Versions 2.007 -- 2.010
      9.1-7 Version 2.12
    9.2 What needs doing next?
  
  
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