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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 $G\le S\sb n:$ (1) given Sylow p-subgroups $P\sb 1$ and $P\sb 2$ of G, find $g\in G$ conjugating $P\sb 1$ to $P\sb 2$; (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 $N\triangleleft G$ with $(\vert N\vert,\vert G/N\vert)=1$ and complements $H\sb 1$ and $H\sb 2$ to N, find $g\in G$ conjugating $H\sb 1$ to $H\sb 2$; (5) given $N\triangleleft G$ with $(\vert N\vert,\vert G/N\vert)=1$, 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 {\it 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

20D20Sylow subgroups of finite groups, Sylow properties, $\pi$-groups, $\pi$-structure
20-04Machine computation, programs (group theory)
68Q25Analysis of algorithms and problem complexity
20B35Subgroups of symmetric groups
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