On the average character degree of finite groups

Abstract

We prove that if the average of the degrees of the irreducible characters of a finite group $G$ is less than 16/5, then $G$ is solvable. This solves a conjecture of I.M. Isaacs, M. Loukaki, and the first author. We discuss related questions.

The first version is revised based on the referee's report. To appear in Bull. Lond. Math. Soc