Working with Finite Groups

Abstract

Two common ways to describe groups are to present them by generators and relations or as automorphism groups of algebraic, geometric or combinatorial structures. (Think of linear groups acting on vector spaces, symmetry groups of regular polytopes, Galois groups etc.) An automorphism group of such a structure may also be considered to be a subgroup of the group of all permutations of the elements of that structure. Automorphism groups can thus be seen as permutation groups. Permutation groups are groups consisting of permutations of a set with composition of permutations as group multiplication. So, for example, we may view linear groups as permutation groups on the set of vectors of the underlying vector space (but this may not be the most efficient approach). The Todd-Coxeter coset enumeration method provides, among other things, a link between groups given by generators and relations on the one hand and permutation groups on the other.