Irreducible characters of even degree and normal Sylow $2$-subgroups

TL;DR

This paper introduces a new invariant related to character degrees and proves that if the average degree of certain irreducible characters is below a threshold, then the group has a normal Sylow 2-subgroup, extending classical results.

Contribution

It proposes a novel invariant involving character degrees and establishes new criteria for the existence of normal Sylow 2-subgroups based on average character degrees.

Findings

If the average degree of linear and even-degree characters is less than 4/3, then the group has a normal Sylow 2-subgroup.

Analogous results hold for real-valued and strongly real characters.

The results improve upon earlier theorems related to the Itô-Michler theorem.

Abstract

The classical It\^o-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group $G$ is coprime to a given prime $p$, then $G$ has a normal Sylow $p$-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of $G$ is less than $4/3$ then $G$ has a normal Sylow $2$-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the It\^o-Michler theorem.