Computing in permutation and matrix groups III: Sylow subgroups

Abstract

The ability of construct the Sylow subgroups of a large finite permutation or matrix group is fundamentalto a number of important algorithms for analyzing the abstract structure of such a group. In this paper we describe a backtrack algorithm which constructs a Sylow subgroup by successive cyclic extensions, starting with a cyclic subgroup of p -power order. The algorithm is capable of finding Sylow p -subgroups of order up to p 10 , for small primes p , in permutation groups having a degree of several hundred.