XMod
  
  
                     Crossed modules and cat1-groups in GAP
  
  
                                  Version 2.12
  
  
                                 November 2008
  
  
                                   Murat Alp
  
                                 Chris Wensley
  
  
  
  Murat Alp
      Email:    mailto:malp@dumlupinar.edu.tr
      Address:  Dumlupinar Universitesi,
                Fen-Edebiyat Fakultesi, Matematik Bolumu
                Merkez Kampus, Kutahya, Turkey.
  
  
  Chris Wensley
      Email:    mailto:c.d.wensley@bangor.ac.uk
      Homepage: http://www.bangor.ac.uk/~mas023/
      Address:  School of Computer Science, Bangor University,
                Dean Street, Bangor, Gwynedd, LL57 1UT, U.K.
  
  
  
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  Abstract
  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).
  
  XMod  was  originally  implemented in 1997 using the GAP3 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 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.
  
  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 mailto:c.d.wensley@bangor.ac.uk.
  
  
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  Copyright
  © 1997-2008 Murat Alp and Chris Wensley
  
  
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  Acknowledgements
  This  xmod  package  is released under the GNU General Public License (GPL).
  This  file is part of xmod, though as documentation it is released under the
  GNU            Free           Documentation           License           (see
  http://www.gnu.org/licenses/licenses.html#FDL).
  
  xmod  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.
  
  xmod  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
  xmod;  if not, write to the Free Software Foundation, Inc., 59 Temple Place,
  Suite 330, Boston, MA 02111-1307 USA.
  
  For more details, see http://www.fsf.org/licenses/gpl.html.
  
  This  documentation  was  prepared with the GAPDoc package of Frank L\"ubeck
  and Max Neunh\"offer.
  
  The  first  author  wishes to acknowledge support from Dumlupinar University
  and the Turkish government.
  
  
  -------------------------------------------------------
  
  
  Contents (XMod)
  
  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 GAP 3
      9.1-2 Version 2
      9.1-3 Version 2.001 for GAP 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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