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

The minimal base size for a p-solvable linear group

Let $V$ be a finite vector space over a finite field of order $q$ and of characteristic $p$. Let $Gleq GL(V)$ be a $p$-solvable completely reducible linear group. Then there exists a base for $G$ on $V$ of size at most $2$ unless $q leq 4$ in which case there exists a base of size at most $3$. The first statement extends a recent result of Halasi and Podoski and the second statement generalizes a theorem of Seress. An extension of a theorem of Palfy and Wolf is also given.