Computational Complexity in Finite Groups

We survey recent results on the asymptotic complexity of some of the fundamental computational tasks in finite groups in a variety of computational models. A striking recent feature is that techniques motivated by the problems of the more abstract models (nondeterminism, extreme parallelization) have turned out to provide powerful tools in the design of surprisingly efficient algorithms on realistic models (e.g. a nearly linear time membership test for permutation groups with a small base). The techniques involve a combination of elementary combinatorial results on finite groups, some classical elementary group theory, and the extensive use of certain consequences of the classification of finite simple groups (CFSG). Most of the recent work surveyed is due to E. M. Luks, G. Cooperman, L. Finkelstein, A. Seress, E. Szemerédi, and the author.