On a conjecture of Gluck

Abstract

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 $|G:F(G)|\le b(G{)}^2$. 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.

16 pages

References in corpus (2)

• The Largest Irreducible Representations of Simple Groups
• The minimal base size for a p-solvable linear group