For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set in any copy of the group, our presentations can be used to verify claimed isomorphisms between representations of the classical group. The presentations are available in Magma.
 We build a theory of black box groups, and apply it to matrix groups over finite fields. Elements of a black box group are encoded by strings of uniform length and group operations are performd by an oracle. Subgroups are given by a list of generators. We prove that for such subgroups, membership and divisor of the order are in NPB. (B is the group box oracle.) Under a plausible mathematical hypothesis on short presentations of finite simple groups, nom membership and exaact order will also be in NPB and thus in NPB ∩ NPB. We give a presentation of length O(log2|G|) for the groups G ≅ PSU3(q). This result has applications in recent algorithms to compute the structure of permutation groups and matrix groups. We conjecture that every finite groupGhas a short presentation (in terms of generators and relations) in the sense that the totallengthof 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 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((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.