Sylow's Theorem in Polynomial Time

Given a set r of permutations of an n-set, let G be the group of permutations generated by f. If p is a prime, a Sylow p-subgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow p-subgroup exists, and that for any two Sylow p-subgroups PI, P, of G there is an element go G such that Pz = g-‘PI g. We present polynomial-time algorithms that find (generators for) a Sylow p-subgroup of G, and that find ge G such that P, = g-‘P, g whenever (generators for) two Sylow p-subgroups PI, Pz are given. These algorithms involve the classification of all tinite simple groups.