Short Presentations for Finite Groups Original Research Article

We conjecture that every finite groupGhas a short presentation (in terms of generators and relations) in the sense that the totallengthof the relations is (logG)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 ofG, assuming in the case of Lie type simple groups overGF(pm) that an irreducible polynomialfof degreemoverGF(p) and a primitive root ofGF(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 groupsPSU(3, q) = 2A2(q), the Suzuki groupsSz(q) = 2B2(q), and the Ree groupsR(q) = 2G2(q). In particular,all finite groups G without composition factors of these types have presentations of length O((logG)3). For groups of Lie type (normal or twisted) of rank ≥ 2, we use a reduced version of the Curtis–Steinberg–Tits presentation.