Indices of centralizers for Hall-subgroups of linear groups (Q1300133)

For a nontrivial Sylow-\(p\)-subgroup of a solvable transitive permutation group \(G\) of degree \(n\), we show that the number of \(P\)-orbits is at most \(2n/(p+1)\). This in turn is used to prove that whenever \(G\) is a solvable irreducible linear group of degree \(n\) with Sylow-\(p\)-subgroup \(P>1\), then the dimension of the centralizer of \(P\) is at most \(2n/(p+1)\). This generalizes (and provides alternate proofs of) results of Isaacs and Navarro. It is also used to affirmatively answer a question of Pérez Monasor and Iranzo Aznar regarding indices of centralizers in coprime operator groups.