Suppose that P is a Sylow-p-subgroup of a solvable group G.If G is a transitive permutation group of degree n,then the number of P-orbits is at most 2n/(p+1).This is used to prove that if G is a faithful irreductible linear group fo degree n, then the dimension of the centralizer of P is at most 2n/(p+1).The latter result generalizes results fo Isaacs and Navarro and is alos used to affirmatively answer a question of Monasur and Iranzo regarding indices of centralizers in coprime operator groups.
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