On a Conjecture of Gluck

Let \(\mathbf {F}(G)\) and \(b(G)\) respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group \(G\). A well-known conjecture of D. Gluck claims that if \(G\) is solvable then \(|G:\mathbf {F}(G)|\le b(G)^{2}\). We confirm this conjecture in the case where \(|\mathbf {F}(G)|\) is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure.