Minimal group presentations : a computational approach

Abstract

The method of describing a group by means of generators and relations is an old one. A question which arises when using this method is what is the minimum number of relations required to describe a given group. Schur (1907) provided a lower bound for the number of relations required, in terms of a group invariant known as the Schur multiplicator. It would be interesting to know which finite groups have a presentation achieving the Schur bound on the number of relations required, but this remains an open question. It is known that the Schur bound is not achievable for some finite groups. We consider the problem of finding a minimal presentations for a number of finite groups. In Chapter Three and Appendix A we give minimal presentations for the groups of order less than or equal to 84 and minimal presentations for some families of groups having composition length less than five and order greater than 84. Some of the techniques for working with finitely presented groups are illustrated by proving that the groups defined by some families of deficiency zero presentations are finite. In Chapter Four, we give presentations for several finite groups having soluble length five and six, and deficiency zero presentations for two infinite families of finite groups, one family having soluble length six, and the other having soluble length five; we also give a deficiency one presentation for a finite preimage of a group having soluble length seven. Finally, in Chapter Five we give some minimal presentations for some quasi-simple groups.