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Finding Sylow normalizers in polynomial time. (English) Zbl 0731.20005

Let G be a subgroup of \(S_ n\), the symmetric group on n points. In continuation of his previous articles on polynomial-time algorithms for finding Sylow p-subgroups of G the author shows in the present paper that there exists also a polynomial-time algorithm for given prime p which finds the normalizer \(N_ G(P)\) of a Sylow p-subgroup P of G. This is done by polynomial-time reducing the problem to an algorithm called SIMPLENORMALIZER. It deals with simple factors of G, also in polynomial time. In particular, in the case where G is solvable the author generalizes this result to Hall \(\pi\)-subgroups for a set of primes \(\pi\).

MSC:

20B40 Computational methods (permutation groups) (MSC2010)
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20B35 Subgroups of symmetric groups
68Q25 Analysis of algorithms and problem complexity
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