Short presentations for finite groups

We conjecture that every finite group G has a short presentation (in terms of generators and relations) in the sense that the rotal length of the relations is (log\G\)(0(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 CF(p(m)) that an irreducible polynomial f of degree m over GF(p) and a primitive root of GF(p(m)) 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) = (2)A(2)(q), the Suzuki groups Sz(q) = B-2(2)(q), and the Ree groups R(q) = (2)G(2)(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 greater than or equal to 2, we use a reduced version of the Curtis-Steinberg-Tits presentation. (C) 1997 Academic Press.