Polynomial-time theory of matrix groups

We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic problems (factoring and discrete log); in fact, constructive membership testing in the case of 1 × 1 matrices is precisely the discrete log problem. So the reasonable question is whether these problems are solvable in randomized polynomial time using number theory oracles. Building on 25 years of work, including remarkable recent developments by several groups of authors, we are now able to determine the order of a matrix group over a finite field of odd characteristic, and to perform constructive membership testing in such groups, in randomized polynomial time, using oracles for factoring and discrete log. One of the new ingredients of this result is the following. A group is called semisimple if it has no abelian normal subgroups. For matrix groups over finite fields, we show that the order of the largest semisimple quotient can be determined in randomized polynomial time (no number theory oracles required and no restriction on parity). As a by-product, we obtain a natural problem that belongs to BPP and is not known to belong either to RP or to coRP. No such problem outside the area of matrix groups appears to be known. The problem is the decision version of the above: Given a list A of nonsingular d × d matrices over a finite field and an integer N, does the group generated by A have a semisimple quotient of order > N? We also make progress in the area of constructive recognition of simple groups, with the corollary that for a large class of matrix groups, our algorithms become Las Vegas.