P-Singular characters and normal Sylow p-subgroups

Abstract

Let [Formula: see text] be a finite group and [Formula: see text] a prime. We denote by [Formula: see text] the set of irreducible complex characters of [Formula: see text] whose degrees are linear or divisible by [Formula: see text], and we write [Formula: see text] to denote the ratio of the sum of squares of irreducible character degrees in [Formula: see text] to the sum of irreducible character degrees in [Formula: see text]. The Itô–Michler Theorem on character degrees states that [Formula: see text] if and only if [Formula: see text] has a normal abelian Sylow [Formula: see text]-subgroup. We generalize this theorem as follows: if [Formula: see text], then [Formula: see text] has a normal Sylow [Formula: see text]-subgroup.