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Residue-Class-Wise Affine Groups


6 December 2022

Stefan Kohl


RCWA is a package for GAP 4. It provides implementations of algorithms and methods for computing in certain infinite permutation groups acting on the set of integers. This package can be used to investigate the following types of groups and many more:

With the help of this package, the author has found a countable simple group which is generated by involutions interchanging disjoint residue classes of ℤ and which all the above groups embed into -- see [Koh10].


© 2003 - 2018 by Stefan Kohl.

RCWA 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.

RCWA is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

For a copy of the GNU General Public License, see the file GPL in the etc directory of the GAP distribution or see


I am grateful to John P. McDermott for the discovery that the group discussed in Section 7.1 is isomorphic to Thompson's Group V in July 2008, and to Laurent Bartholdi for his hint on how to construct wreath products of residue-class-wise affine groups with (ℤ,+) in April 2006. Further, I thank Bettina Eick for communicating this package and for her valuable suggestions on its manual in the time before its first public release in April 2005. Last but not least I thank the two anonymous referees for their constructive criticism and their helpful suggestions.


1 About the RCWA Package
2 Residue-Class-Wise Affine Mappings
3 Residue-Class-Wise Affine Groups
4 Residue-Class-Wise Affine Monoids
5 Residue-Class-Wise Affine Mappings, Groups and Monoids over ℤ^2
6 Databases of Residue-Class-Wise Affine Groups and -Mappings
7 Examples
8 The Algorithms Implemented in RCWA
9 Installation and Auxiliary Functions

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