Addendum to: “An elementary introduction to coset table methods in computational group theory”

Abstract

Listen

Translate article

Even after 25 years the article [30] by Joachim Neubuser remains the first source to which all three of us refer those who want to find out about the use of coset tables for studying groups. Our view is confirmed by the 14 Reference Citations from 1998 to 2005 which MathSciNet [1] reveals for this article. Here we loosely follow the structure of the original article and provide some updates on the area (oriented towards our own interests). First we point out that two newer books [35, 22] include comprehensive details on coset enumeration and related topics in works which are much broader studies. They give excellent coverage of the areas addressed in this article and, further, provide much additional material. They also provide some alternative points of view and many references (as do the other materials cited here). One of Neubuser’s aims in writing his survey was to provide a unified view on coset table methods in computational group theory. He addressed the way coset table concepts were developed, implemented and used. In [22] Derek Holt follows the same kind of approach, including a long chapter “Coset Enumeration” and a shorter one “Presentations for Given Groups”. Charles Sims in [35] focuses on finitely presented groups and he takes a perspective significantly based on some fundamental methods from theoretical computer science, namely automata theory and formal languages. He includes three chapters specifically relevant to coset table methods: “Coset enumeration”; “The Reidemeister-Schreier procedure”; and “Generalized automata”; with some extra implementation issues covered in an Appendix. He concludes his coset enumeration chapter with a section which points out that the Knuth-Bendix procedure can sometimes be used more effectively to enumerate cosets than Todd-Coxeter methods. Among the available computer implementations of coset enumeration procedures are those in the computer algebra systems GAP [14] and Magma [5] and a standalone program, the ACE coset enumerator [18]. An implementation is also available via quotpic [23], a software package with a nice graphical interface. A particularly useful tool for small-scale experiments with coset enumerations is the Interactive