The average character degree of finite groups and Gluck's conjecture

TL;DR

This paper investigates the relationship between the average degree of irreducible characters and the structure of finite groups, proposing a refined version of Gluck's conjecture based on new bounds and conditions.

Contribution

It establishes bounds on the order of finite groups with trivial solvable radical using average character degree and introduces a refined Gluck's conjecture considering character degrees over linear Fitting subgroup characters.

Findings

Order of certain finite groups is bounded by average character degree

The index of the Fitting subgroup is not generally bounded by average character degree

A refined Gluck's conjecture is proposed based on new bounds

Abstract

We prove that the order of a finite group $G$ with trivial solvable radical is bounded above in terms of ${\rm acd}(G)$, the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded above in terms of ${\rm acd}(G)$, but we show that in certain cases it is bounded in terms of the degrees of the irreducible characters of $G$ that lie over a linear character of the Fitting subgroup. This leads us to propose a refined version of Gluck's conjecture.