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

We prove that the order of a finite group $G$ with trivial solvable radical is bounded above in terms of $\mathrm{a}\mathrm{c}\mathrm{d}(G)$, the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded above in terms of $\mathrm{a}\mathrm{c}\mathrm{d}(G)$, 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.