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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 GS n : (1) given Sylow p-subgroups P 1 and P 2 of G, find gG 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 NG with (|N|,|G/N|)=1 and complements H 1 and H 2 to N, find gG conjugating H 1 to H 2 ; (5) given NG with (|N|,|G/N|)=1, find a complement to N in G. If G is solvable, the analogues of (1), (2), and (3) for π-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:
20D20Sylow subgroups of finite groups, Sylow properties, π-groups, π-structure
20-04Machine computation, programs (group theory)
68Q25Analysis of algorithms and problem complexity
20B35Subgroups of symmetric groups