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.