Computing in permutation and matrix groups III: Sylow subgroups

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 p10, for small primes p, in permutation groups having a degree of several hundred