Computational techniques and the structure of groups in a certain class

Abstract

The Todd-Coxeter algorithm enumerates the cosets of a finitely generated subgroup of finite index in a finitely presented group. The algorithm has been modified to give a presentation of the subgroup in terms of the given generators. In this paper we describe briefly computer programmes for the algorithm and the modified algorithm, and illustrate how the modified algorithm may be extended to subgroups of countable index. We then describe some methods that enable us to obtain more information from a coset enumeration programme when space is limited. To illustrate these techniques we investigate a class of cyclically presented groups. We first state some theorems concerning this class. Using the computational techniques described earlier, we then discuss in some detail the structure of some of the particular groups involved. The paper arises out of joint work with E. F. Robertson at the University of St. Andrews.