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 PΩ(2m+1, q) and PSp(2m, q) in this manner when q is odd and m ≥ 3. These two groups can, however, be distinguished by using an algorithm of Altseimer and Borovik. 2000 Mathematics Subject Classification: Primary 20D06; Secondary: 20-04, 20P05, 68W30. ∗ This research was supported in part by the National Science Foundation. ∗∗ This research was supported in part by grant OTKA T29132, Hungary.