Sylow Subgroups in Parallel

Sylow subgroups are fundamental in the design of asymptotically efficient group-theoretic algorithms, just as they have been in the study of the structure of Ž . finite groups. We present efficient parallel NC algorithms for finding and conjugating Sylow subgroups of permutation groups, as well as for related problems. Polynomial-time solutions to these problems were obtained more than a dozen years ago, exploiting a well-developed polynomial-time library. We replace some of those highly sequential procedures with ones that work through a polylog-length normal series that is a by-product of NC membership-testing. As in previous investigations, we reduce to the base case of simple groups, and handle this by a case-by-case analysis that depends heavily upon the classification of finite simple groups. Q 1999 Academic Press