Permutation Group Algorithms

Computational group theory (CGT) is a subfield of symbolic algebra; it deals with the design, analysis, and implementation of algorithms for manipulating groups. It is an interdisciplinary area between mathematics and computer science. The major areas of CGT are the algorithms for finitely presented groups, polycyclic and finite solvable groups, permutation groups, matrix groups, and representation theory.

The topic of this book is the third of these areas. Permutation groups are the oldest type of representations of groups; in fact, the work of Galois on permutation groups, which is generally considered as the start of group theory as a separate branch of mathematics, preceded the abstract definition of groups by about a half a century. Algorithmic questions permeated permutation group theory from its inception. Galois group computations, and the related problem of determining all transitive permutation groups of a given degree, are still active areas of research (see [Hulpke, 1996]). Mathieu's constructions of his simple groups also involved serious computations.

Nowadays, permutation group algorithms are among the best developed parts of CGT, and we can handle groups of degree in the hundreds of thousands. The basic ideas for handling permutation groups appeared in [Sims, 1970, 1971a]; even today, Sims's methods are at the heart of most of the algorithms.