Finding Sylow Normalizers in Polynomial Time

Given a set Γ of permutations of an n-set, let G be the group of permutations generated by Γ. If p is any prime, it is known that a Sylow p-subgroup P of G can be found in polynomial time. We show that the normalizer of P can also be found in polynomial time. In particular, given two Sylow p-subgroups of G, all elements conjugating one to the other can be found (as a coset of the normalizer of one of the Sylow p-subgroups). Analogous results are obtained in the case of Hall subgroups of solvable groups.

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