Presentations of linear groups

Abstract

If d(M) denotes the rank of the Schur multiplicator of a finite group G, then a group is efficient if -def G = d(M). Efficient presentations of the simple groups PSL(2,p), p an odd prime > 3, were obtained by J.G. Sunday.This raised the question of whether or not all finite simple groups are efficient.

In this thesis, we investigate the deficiency of the groups PSL(2,pⁿ). J.A. Todd gave presentations for PSL(2,pⁿ) which use large numbers of generators and relations. Starting with these, we obtain, at best, deficiency -1 presentations for PSL(2,2ⁿ) (= SL(2,2ⁿ)) and deficiency -6 presentations for PSL(2,pⁿ), p an odd prime. If pⁿ = -1(mod 4), the latter can be reduced to a deficiency -4 presentation. Efficient presentations for PSL(2,25), PSL(2,27) and PSL(2,49) are obtained.

The Behr-Mennicke presentation for PSL(2,p) is one of the most fundamental in the sense that it forms the basis for others, such as those given by Sunday, Zassenhaus and Sidki. Behr and Mennicke derived their presentation indirectly, and it would be desirable to have a more direct proof. The groups G[sub]p(a) are defined as

< U, R, S | U³ = (UR)² = (US)² = Sᵖ = Rᵗ = (SaRU)³= 1, Sᵃ²R = RS >

where a ε GF(p)* and a²ᵗ = 1 (mod p) . We show that G[sub]p (2) is isomorphic with the Behr-Mennicke presentation for PSL(2,p), p > 3. Conditions are found to discover when Gp (a) is isomorphic with PSL(2,p) and, under these conditions, this provides a direct proof of the Behr-Mennicke presentations. For any odd positive integer m, we show that the groups SL(2,ℤ (m)) and PSL(2,ℤ(m)) are efficient.