Indices of centralizers for Hall-subgroups of linear groups

Abstract

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 irreducible linear group of degree $n$, then the dimension of the centralizer of $P$ is at most $2n/(p + 1)$. The latter result generalizes results of Isaacs and Navarro and is also used to affirmatively answer a question ofMonasur and Iranzo regarding indices of centralizers in coprime operator groups.