Physical Sciences and Mathematics Commons^™

On A Conjecture Of Gluck, James Cossey, Zoltan Halasi, Attila Maroti, Hung Nuguyen

James P. Cossey

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.