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Crossed Modules and Cat1-Groups


23 January 2024

Chris Wensley

Murat Alp
Prof. Dr. M. Alp
Ömer Halisdemir University
Art and Science Faculty
Mathematics Department

Alper Odabas
Dr. A. Odabas
Osmangazi University
Arts and Sciences Faculty
Department of Mathematics and Computer Science

Enver Onder Uslu


The XMod package provides functions for computation with

XMod was originally implemented in 1996 using the GAP3 language, when the second author was studying for a Ph.D. [Alp97] at 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.

In October 2015 functions for computing isoclinism classes of crossed modules, written by Alper Odabaş and Enver Uslu, were added. These are contained in Chapter 4, and are described in detail in the paper [IOU16].

Bug reports, suggestions and comments are, of course, welcome. Please submit an issue at or send an email to the first author at


© 1996-2024, Chris Wensley et al.

The XMod package 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.


This documentation was prepared using the GAPDoc [LN17] and AutoDoc [GH17] packages.

The procedure used to produce new releases uses the package GitHubPagesForGAP [Hor17] and the package ReleaseTools.


1 Introduction
2 2d-groups : crossed modules and cat\(^1\)-groups
3 2d-mappings
4 Isoclinism of groups and crossed modules
5 Whitehead group of a crossed module
6 Actors of 2d-groups
7 Induced constructions
8 Crossed squares and Cat\(^2\)-groups
9 Crossed cubes and Cat\(^3\)-groups
10 Crossed modules of groupoids
11 Double Groupoids
12 Applications
13 Interaction with HAP
14 Utility functions
15 Development history

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