Symmetric presentations and orthogonal groups

Summary

Abstract

In this paper we examine series of finite presentations which are invariant under the full symmetric group acting on the set of generators. Evidence from computational experiments reveals a remarkable tendency for the groups in these series to be closely related to the orthogonal groups. We examine cases of finite groups in such series and look in detail at an infinite group with such a presentation. We prove some theoretical results about 3-generator symmetric presentations and make a number of conjectures regarding n-generator symmetric presentations.

Introduction

Symmetric presentations have been much studied over a long period; see for example [1, 9, 10, 14, 23], which focus on symmetric relations. An alternative approach, directed towards symmetric generating sets, is taken by Curtis [11, 12, 13].

Suppose G is a finite 2-generator group with generators x₁,x₂ and an automorphism θ with x₁θ = x₂ θ and θ² = 1. Then G has a symmetric presentation

Note that any finite non-abelian simple group has such a symmetric presentation. For, if G is a finite non-abelian simple group, then G = 〈a, b〉 where a² = 1 [19]. Now consider H = 〈b, b^a〉 ≤ G. Either G = H or |G : H| = 2. But G is simple, so G = H = 〈b, b^a〉 and hence G has a symmetric presentation.

Recently Miklos Abert (unpublished) has generalized the results of [14] and conjectures that, given any non-trivial group G, then G has a presentation which, when symmetrized, presents a non-trivial image of G. The above comment on symmetric presentations for simple groups is a consequence of the results of Abert.