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Calculations with finite groupoids and their homomorphisms


23 January 2024

Emma J. Moore

Chris Wensley


The groupoids package provides functions for computation with groupoids (categories with every arrow invertible) and their morphisms; for graphs of groups, and graphs of groupoids. The most basic structure introduced is that of magma with objects, followed by semigroup with objects, then monoid with objects and finally groupoid which is a group with objects.

It provides normal forms for Free Products with Amalgamation and for HNN-extensions when the initial groups have rewrite systems and the subgroups have finite index. This is described in Section 6.2. It is planned to move this section to a new package Rewriting in time for version 4.11 of GAP.

The groupoids package was originally implemented in 2000 (as GraphGpd) when the first author was studying for a Ph.D. in Bangor.

The package was then renamed Gpd and version 1.07 was released in July 2011, ready for GAP 4.5.

Gpd became an accepted GAP package in May 2015.

In April 2017 the package was renamed again, as groupoids.

Recent versions implement many of the constructions described in the paper [AW10] for automorphisms of groupoids.

Bug reports, comments, suggestions for additional features, and offers to implement some of these, will all be very welcome.

Please submit any issues at or send an email to the second author at


© 2000-2024, Emma Moore and Chris Wensley.

The groupoids 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 Many-object structures
3 Mappings of many-object structures
4 Groupoids
5 Homomorphisms of Groupoids
6 Graphs of Groups and Groupoids
7 Double Groupoids
8 Technical Notes
9 Development History

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