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

Abstract We prove that the order of a finite group 𝐺 with trivial solvable radical is bounded above in terms of acd ⁡ ( G ) \operatorname{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 acd ⁡ ( G ) \operatorname{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.