Short Presentations for Finite Groups

We conjecture that every nite group G has a short presentation (in terms of generators and relations) in the sense that the total length of the relations is (log jGj) O(1). We show that it suuces to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for nite 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(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 nite 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) = 2 B 2 (q), and the Ree groups R(q) = 2 G 2 (q). In particular, all nite groups G without composition factors of these types have presentations of length O((log jGj) 3). For groups of Lie type (normal or twisted) of rank 2, we use a reduced version of the Curtis-Steinberg-Tits presentation.