Irreducible characters of even degree and normal Sylow 2-subgroups

Abstract

Abstract. The classical Itô-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 Itô-Michler theorem.

Variations on average character degrees and solvability

“…By [10,Corollary 1.4], if G is non-solvable, then G admits a real irreducible character of even degree. This shows that if F ∈ {R, C} and acd F,even (G) = 0, then G is solvable.…”

On the average character degree of finite groups

“…Similarly, we write acd(G) = χ∈Irr(G) χ( 1) k(G) to denote the average degree of the irreducible characters of G. This invariant was introduced in [10], motivated by a conjecture of K. Magaard and H. Tong-Viet [14]. Since then, acd(G) has been studied in a number of papers (see, for instance, [17,19,13,7,6,8]), but several fundamental questions remain open. The previously done work shows that working with the average character degree tends to be more difficult than working with the average class size.…”