Development history

This chapter, which is intended to contain details of the major changes to the package as it develops, was first created in April 2002. Details of the changes from &XMod;~1 to &XMod;~2.001 are far from complete.

The inspiration for this package was the need, in the mid-1990's, to calculate induced crossed modules (see , , ). &GAP; was chosen over other computational group theory systems because the code was freely available, and it was possible to modify the Tietze transformation code so as to record the images of the original generators of a presentation as words in the simplified presentation. (These modifications are now a standard part of the Tietze transformation package in &GAP;.)

Changes from version to version
Version 1 for &GAP;~3
version 1 for &GAP;~3

The first version of &XMod; became an accepted package for &GAP;~3.4.3 in December 1996.
Version 2

Conversion of &XMod;~1 from &GAP;~3.4.3 to the new &GAP; syntax began soon after &GAP;~4 was released, and had a lengthy gestation. The new &GAP; syntax encouraged a re-naming of many of the function names. An early decision was to introduce generic names 2dObject for (pre-)crossed modules and (pre-)cat1-groups, and 2dMapping for the various types of morphism. This allows 3dObject to be used in future for crossed squares and cat2-groups, and 3dMapping for their morphisms. A generic name for derivations and sections is also required, and Up2dMapping is currently used.
Version 2.001 for &GAP;~4
version 2.001 for &GAP;~4

This was the first version of &XMod; for &GAP;~4, completed in April 2002 in a rush to catch the release of &GAP;~4.3. Functions for actors and induced crossed modules were not included, nor many of the functions for derivations and sections, for example InnerDerivation. During the ten days prior to the release, the main changes made were: Generic name UpMapping chosen for derivations and sections (now changed to Up2dMapping). File names changed to obj2.gd, map2.gi, up2.tex, etc. Added alternative methods for IsomorphismPermGroup for 2dObjects (see also 9.1.4). Sorted a problem with fixing the generating set for R when used to define derivations. The (old) code uses an fp-group version of R and checks that all the relators map by chi to 1. Unfortunately, IsomorphismFpGroup sometimes permutes the order of the R-generators, with unfortunate effects. The fix is to use IsomorphismFpGroupByGenerators which returns the images of the generators specified in the function call. We have also used genR := StrongGeneratorsStabChain( StabChain( rng ) ); throughout to specify the generators of R. Operation XModMorphism renamed as XModMorphismByHoms, and a new global function XModMorphism introduced (ditto for other 2dMappings). Now using \chi_1 \star \chi_2 for Whitehead multiplication {\bf on the right}, with CompositeDerivation still giving multiplication {\bf on the left}. This means that the second axiom for derivations and for sections has changed -- see Chapter 4.
Induced crossed modules

During the period May 20th - May 27th 2002 converted induce.g to induce.gd and induce.gi (later renamed induce2.gd, induce2.gi), at least as regards induced crossed modules. (Induced cat1-groups may be convereted one day.) This involved the following. Converted combinatorial functions -- DistinctRepresentatives, CommonRepresentatives, CommonTransversal and IsCommonTransversal. Converted Tietze modification functions TzCommutatorPair, TzPartition and FactorsPresentation. Introduced global functions IsomorphismPermObject, IsomorphismFpObject, and IsomorphismPcObject which call IsomorphismPermGroup etc. when the object is a group. Added functions IsomorphismPermPreXMod, IsomorphismPermPreCat1, etc. to be called when the object is a 2d-object. Added IsomorphismXModByNormalSubgroup which applies when the boundary of the xmod is injective. Added PreXModIsomorphismByIsomorphisms (we also need a similar function PreCat1IsomorphismByIsomorphisms) where the data consists of a crossed module, an isomorphism of the source, and an isomorphism of the range. Changed RModule to AbelianModule.
Versions 2.002 -- 2.006

Version 2.002 was prepared for the 4.4 release at the end of January 2004, and so required a PackageInfo.g file.

Version 2.003 of February 28th 2004 just fixed some file protections.

Version 2.004 of April 14th 2004 gave a new email address for Murat Alp and added the Cat1Select functionality of version 1 to the Cat1 function (see also version 2.007).

Version 2.005 of April 16th 2004 moved the example files from tst/test&uscore;i.g to examples/example&uscore;i.g, and converted testmanual.g to a proper test file tst/xmod&uscore;manual.tst.

Changes made include the following. Replaced OperationHomomorphism by ActionHomomorphism -- a general &GAP;4.4 change. Finished replacing RModule by AbelianModule. Renamed UpMapping as Up2dMapping. Added MappingGeneratorsImages and InverseGeneralMapping for a 2dMapping. Converted the actor crossed module functions from the 3.4.4 version, including AutomorphismPermGroup for a crossed module, WhiteheadXMod, NorrieXMod, LueXMod, ActorXMod, Centre of a crossed module, InnerMorphism and InnerActorXMod. Added SmallerDegreePermPreXMod after discovering, in the library, the very useful SmallerDegreePermutationRepresentation.

Version 2.006 of September 4th 2004 changed morphism functions to return fail when invalid data is supplied, rather than calling Error, and fixed a bug in XModByGroupOfAutomorphisms.
Versions 2.007 -- 2.008

This version contains changes made between September 2004 and October 2006. Added basic functions for crossed squares, considered as 3dObjects with crossed pairings, and their morphisms. Groups with two normal subgroups, and the actor of a crossed module, provide standard examples of crossed squares. (Cat2-groups are not yet implemented.) Converted the documentation to the format of the &GAPDoc; package, and added the function XModBuildManual() to the util.gi file. Improved AutomorphismPermGroup for crossed modules, and introduced a special method for conjugation crosseed modules. Substantial revisons made to XModByCentralExtension, NorrieXMod, LueXMod, ActorXMod, and InclusionInducedXModByCopower. Reintroduced the Cat1Select operation. The only reason for releasing version 2.008 was to fix broken links in PackageInfo.g.
What needs doing next?
Speed up the calculation of Whitehead groups.
Add more functions for 3dObjects and implement cat2-groups.
Add interaction with `IdRel`, `XRes`, and `natp`.
Need InverseGeneralMapping for morphisms.
Need more features for FpXMods, PcXMods, etc.
Implement actions of a crossed module.
Implement FreeXMods.
Implement an operation Isomorphism2dObjects.
Allow the construction of a group of morphisms of crossed modules.
Complete the conversion from Version 1 of the calculation of sections using EndoClasses.