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This is a page on GAP 3, which is still available, but no longer supported. The present version is GAP 4  (See  Status of GAP 3).

GAP 3 Share Package "xmod"

The XMod package

Share package since release 3.4, about December 1996, communicated by Derek Holt.
Corresponding GAP 4 package: GAP 4 package "xmod".


Chris Wensley and Murat Alp.


Language: GAP 3
Operating system: Any
Current GAP 3 version: 1.3.1 (in the 3.4.4 distribution)


This package enables construction of and computation with objects within the equivalent categories of crossed modules and cat1-groups, where the groups involved are finite permutation groups of moderately small order.

A crossed module consists of two groups S and R, together with a homomorphism from S to R which essentially commutes with conjugation within S and R. Functions are provided for each of the standard constructions of crossed modules, and for computing with sub- and quotient-structures and homomorphisms.

A cat1-group consists of a group G together with two homomorphisms from G to G satisfying certain conditions. It was shown by Loday that there is a natural one-one correspondence between cat1-groups and crossed modules.

An important notion is a derivation of a crossed module, which is a map from R to S satisfying certain conditions. It is possible to compute all (or all regular) derivations of a crossed module. There are also functions for computing the actor of a crossed module X, which is a crossed module whose range is the automorphism group of X.


An XMod manual is given in chapter 73 of the GAP 3 manual.

Contact addresses

Chris Wensley
School of Informatics
University of Wales
Bangor, Gwynedd, LL57 1UT

Murat Alp