On presentation of PSL (2, pⁿ)

Abstract

We give presentations for the groups PSL(2,p"), p prime, which show that the deficiency of these groups is bounded below. In particular, for p = 2 where SL(2,2") = PSL(2,2"), we show that these groups have deficiency greater than or equal to -2 . We give deficiency -1 presentations for direct products of SL(2,2"') for coprime n,-. Certain new efficient presentations are given for certain cases of the groups considered.

Presentations on standard generators for classical groups

“…We now prove the first relation in (7). An instance with i = 1, j = 2, and α = s = 1 is given by relation (c).…”

“…Presentations for SL(2, q) for e > 1. We use presentations from [7] rewritten on the generating set {τ = τ (1), δ, U }. A complication arises from the fact that τ (α) δ = τ (αω 2 ), so τ (ω i ) is treated differently according to the parity of i.…”

“…This is essentially due to Campbell, Robertson, and Williams [7,Theorem 2.2]. We have replaced their generators w, x, y, z by U τ −1 , τ, τ 1 , and τ δ −1 .…”

“…where r = (q + 1)/4 if 1 + ω ∈ GF(q) 2 , and r = (q − 3)/4 otherwise. This is essentially [7,Theorem 2.4]. We have replaced the relation that translates to (τ δ −1 ) (q−1)/2 = 1 by the simpler relation δ (q−1)/2 = 1, as in Theorem 3.1.…”

The Dehn function of Baumslag’s metabelian group

“…The notation and techniques used in our next lemma are similar to those in Section 4.3 of [20], which in turn draws on [9]. It concerns the impact of the relations a s = aa t and a s = a t a in .…”