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### 9. 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 [RC95], [RC96], [RC03]). 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.)

#### 9.1 Changes from version to version

##### 9.1-1 Version 1 for GAP~3

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

##### 9.1-2 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.

##### 9.1-3 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.

##### 9.1-4 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`.

##### 9.1-5 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_i.g` to `examples/example_i.g`, and converted `testmanual.g` to a proper test file `tst/xmod_manual.tst`.

Changes made include the following.

- Replaced `OperationHomomorphism` by `ActionHomomorphism` -- a general GAP4.4 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`.

##### 9.1-6 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`.

#### 9.2 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`.

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