A Presentation of the Groups PSL(2, p) with Three Defining Relations

Abstract

H. Behr and J. Mennicke (1) have proven that the group PSL(2, p) can be presented by the following system of generators and relations:1From this presentation, it follows that the three relations2for the same generators S and T suffice if p > 3, p ≠ 17. If p = 3, it is well known that the relations S3 = 1, T2 = 1, and (ST)3 = 1 define PSL(2, 3). For p = 2, the relations S3 = 1, T2 = 1, and (ST)2 = 1 define PSL(2, 2). For p = 17, the three relations3will suffice.Indeed, the group G, with generators S, T and defining relations (2), contains the subgroup 〈Sp〉 in its centre.