## 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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