Short presentations for finite groups

Abstract

We conjecture that every finite group G has a short presentation (in terms of generators and relations) in the sense that the total length of the relations is (log|G|)O(1). We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name of G, assuming in the case of Lie type simple groups over GF(pm) that an irreducible polynomial f of degree m over GF(p) and a primitive root of GF(pm) are given. We verify this (stronger) conjecture for all finite simple groups except for the three families of rank 1 twisted groups: we do not handle the unitary groups PSU(3, q) = 2A2(q), the Suzuki groups Sz(q) = 2B2(q), and the Ree groups R(q) = 2G2(q). In particular, all finite groups G without composition factors of these types have presentations of length O((log|G|)3). For groups of Lie type (normal or twisted) of rank ≥ 2, we use a reduced version of the Curtis-Steinberg-Tits presentation. © 1997 Academic Press.