Presentations of finite simple groups: a computational approach

Abstract

Abstract. All finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2 G 2 (q), have presentations with at most 49 relations and bit-length O(log n + log q). Moreover, A n and S n have presentations with 3 generators, 7 relations and bitlength O(log n), while SL(n, q) has a presentation with 6 generators, 25 relations and bit-length O(log n + log q).

Presentations on standard generators for classical groups

“…In two remarkable papers, Guralnick et al [13,14] establish much more. We summarise just two of their results.…”

“…Motivated in part by a conjecture of [3], Babai et al [4] provided presentations on a reduced set of Curtis-Steinberg-Tits generators. Guralnick et al [13,14] provide short presentations (defined below) for every finite simple group of Lie type except the Ree groups 2 G 2 (q). In this paper, we focus on the classical groups.…”

“…Rarely do we know explicit words (short or otherwise) to express standard generators in terms of the generators used in [14], or vice versa, so it is not feasible to convert their presentations to ones on standard generators. One significant obstruction is that sometimes generators used there are identified only by specifying properties they must satisfy.…”

Algorithms for matrix groups

“…Guralnick et al [54] prove that A n has a presentation on 3 generators, 4 relations, and length O(log n). Bray et al [19] prove that S n has a presentation of length O(n 2 ) on generators (1, 2) and (1, 2, .…”

Short presentations for alternating and symmetric groups

“…In a more recent paper [11], they show that both A n and S n have presentations with 3 generators, 7 relators, and length O(log n).…”