Presentations of finite simple groups: A quantitative approach

Abstract

There is a constant C 0 C_0 such that all nonabelian finite simple groups of rank n n over F q \mathbb {F}_q , with the possible exception of the Ree groups 2 G 2 ( 3 2 e + 1 ) ^2G_2(3^{2e+1}) , have presentations with at most C 0 C_0 generators and relations and total length at most C 0 ( log ⁡ n + log ⁡ q ) C_0(\log n +\log q) . As a corollary, we deduce a conjecture of Holt: there is a constant C C such that dim ⁡ H 2 ( G , M ) ≤ C dim ⁡ M \dim H^2(G,M) \leq C\dim M for every finite simple group G G , every prime p p and every irreducible F p G {\mathbb F}_p G -module M M .