*(English)*Zbl 0642.20019

Imposing certain restrictions on the composition factors the authors present polynomial time algorithms for solving the following problems for permutation groups $G\le {S}_{n}:$ (1) given Sylow p-subgroups ${P}_{1}$ and ${P}_{2}$ of G, find $g\in G$ conjugating ${P}_{1}$ to ${P}_{2}$; (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 $N\u25c3G$ with $\left(\right|N|,|G/N\left|\right)=1$ and complements ${H}_{1}$ and ${H}_{2}$ to N, find $g\in G$ conjugating ${H}_{1}$ to ${H}_{2}$; (5) given $N\u25c3G$ with $\left(\right|N|,|G/N\left|\right)=1$, find a complement to N in G. If G is solvable, the analogues of (1), (2), and (3) for $\pi $-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.

##### MSC:

20D20 | Sylow subgroups of finite groups, Sylow properties, $\pi $-groups, $\pi $-structure |

20-04 | Machine computation, programs (group theory) |

68Q25 | Analysis of algorithms and problem complexity |

20B35 | Subgroups of symmetric groups |