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Polynomial-time versions of Sylow’s theorem. (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 $N◃G$ with $\left(|N|,|G/N|\right)=1$ and complements ${H}_{1}$ and ${H}_{2}$ to N, find $g\in G$ conjugating ${H}_{1}$ to ${H}_{2}$; (5) given $N◃G$ with $\left(|N|,|G/N|\right)=1$, 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.

Reviewer: P.P.Pálfy

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