Presentations of finite simple groups: a quantitative approach

Abstract

Every nonabelian finite simple group of rank $n$ over a field of size $q$, with the possible exception of the Ree groups ${}^2{G}_2(3^{2e+1})$, has a presentation with a bounded number of generators and relations and total length $O(\log n+\log q)$. As a corollary, we deduce a conjecture of Holt: there is a constant $C$ such that $\dim H^2(G,M)\le C\dim M$ for every finite simple group $G$, every prime $p$ and every irreducible $F_p[G]$-module $M$.

latex 64 pages