A computer application to finite <i>p</i>-groups

Abstract

The first and still the best known computer application to groups exploits coset enumeration and this has been very thoroughly studied; see for instance [2] and [8]. No doubt this is because the algorithm is simple in the sense of programming. The Underlying mathematics is far from simple, touching as it does no logical difficulties akin to the word problem for groups, and this is reflected in the facts that random access to large tables is required and that there is no indication at any stage (for example when storage space is exhausted) whether the algorithm would be completed at any later stage. Efficient computation depends on choosing a subgroup of small index m in the group under examination, for group elements will be represented as permutations of degree m , and the larger m is the more tedious it will be to check properties like orders of group elements. Yet in many cases m may have to be fairly large so that the subgroup is “corefree” i.e. the representation is faithful.