Imposing certain restrictions on the composition factors the authors present polynomial time algorithms for solving the following problems for permutation groups (1) given Sylow p-subgroups and of G, find conjugating to ; (2) find a Sylow p-subgroup of G; (3) given a p-subgroup K of G, find a Sylow p-subgroup of G containing K; (4) given with and complements and to N, find conjugating to ; (5) given with , find a complement to N in G. If G is solvable, the analogues of (1), (2), and (3) for -subgroups are solved as well.
Polynomial time algorithms for these problems in arbitrary permutation groups can be found in a later paper of W. M. Cantor [J. Comput. Syst. Sci. 30, 359-394 (1985; Zbl 0573.20022)], however that version uses the classification of finite simple groups.