On a conjecture of Gluck

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Let 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:F(G)|≤ b(G)². We confirm this conjecture in the case where |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.