Characters of odd degree and Thompson’s character degree theorem

Abstract

For a positive integer m and a finite group G, let

$$\begin{aligned} u_{2'}(G,m)=\frac{\sum _{\chi \in \mathrm{Irr}_{2'}(G)}\chi (1)^{m}}{\sum _{\chi \in \mathrm{Irr}_{2'}(G)}\chi (1)^{m-1}}, \end{aligned}$$

where $\mathrm{Irr}_{2'}(G)$ denotes the set of all complex irreducible characters of G of odd degrees. The Thompson’s theorem on character degrees states that if $u_{2'}(G,m)=1$, then G is 2-nilpotent. In this paper, we prove that if

$$\begin{aligned} u_{2'}(G,m)< \frac{3+3^{m}}{3+3^{m-1}}, \end{aligned}$$

then G is 2-nilpotent. This is a strengthened version of Thompson’s theorem in terms of $u_{2'}(G,m)$.