On a Conjecture of Gluck
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.
Cossey, James P.; Halasi, Zoltan; Maroti, Attila; and Nuguyen, Hung Ngoc, "On a Conjecture of Gluck" (2015). Mathematics Faculty Research. 9.