5 - A procedure for obtaining simplified defining relations for a subgroup

Summary

INTRODUCTION

Given a finitely presented group G and a set of generators for a subgroup H of finite index in G, the Todd-Coxeter algorithm gives a systematic method for determining the index of H. This algorithm has been the subject of much investigation over the last fifteen years. Various computer implementations have been devised to improve its computational efficiency (see, for example, and) and it has also been modified to allow the construction of a set of defining relations for H.

The main disadvantage of this automatic approach to obtaining subgroup presentations is that, even when the index of H is small, the presentations often contain either a large number of generators, many of which are redundant, or many and long relations - or both. (See, for example.) It is possible to improve these presentations by performing a sequence of Tietze transformations (Chapter IV) to simplify the relations and remove redundant generators, but by doing so we may lose control over the subgroup generators: that is, the resulting generating set in the simplified presentation of H may not be equal to the original set of generators. Since, in some investigations, we are looking for a set of defining relations on a specific set of subgroup generators this approach is not always appropriate.