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Published online by Cambridge University Press: 19 May 2010
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, ba〉 ≤ G. Either G = H or |G : H| = 2. But G is simple, so G = H = 〈b, ba〉 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Recently Miklos Abert (unpublished) has generalized the results of [14] and conjectures that, given any non-trivial group G, then G has a presentation which, when symmetrized, presents a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.
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