[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? -------------------------------------------------------