On the average character degree of finite groups

TL;DR

This paper proves that finite groups with an average irreducible character degree less than 16/5 are necessarily solvable, confirming a conjecture and advancing understanding of group structure based on character degrees.

Contribution

It establishes a new threshold for average irreducible character degrees that guarantees group solvability, solving a longstanding conjecture.

Findings

Groups with average character degree < 16/5 are solvable

Confirmed a conjecture by Isaacs, Loukaki, and the first author

Provides insights into the relationship between character degrees and group structure

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.