Black-box recognition of finite simple groups of Lie type by statistics of element orders

Given a black-box group G isomorphic to some finite simple group of Lie type and the characteristic of G, we compute the standard name of G by a Monte Carlo algorithm, The running time is polynomial in the input length and in the time requirement for the group operations in G. The algorithm chooses a relatively small number of (nearly) uniformly distributed random elements of G, and examines the divisibility of the orders of these elements by certain primitive prime divisors. We show that the divisibility statistics determine G, except that we cannot distinguish the groups POmega(2m + 1, q) and PSp(2m, g) in this manner when q is odd and m greater than or equal to 3. These two groups can, however, be distinguished by using an algorithm of Altseimer and Borovik.