p-Singular characters and normal Sylow p-subgroups

Let G be a finite group and p a prime. We denote by [Formula: see text] the set of irreducible complex characters of G whose degrees are linear or divisible by p, 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 G has a normal abelian Sylow p-subgroup. We generalize this theorem as follows: if [Formula: see text], then G has a normal Sylow p-subgroup.