Large character degrees of groups of odd order
A very powerful regular orbit theorem for quasiprimitive modules of groups of odd order is obtained. It is applied to the following conjecture due to Gluck: If G is a solvable group, then \(| G:F(G)| \leq b(G)^ 2\), where \(b(G)=\max\{\chi(1)|\chi\in Irr(G)\}\). Here the conjecture is proven when \(| G|\) is odd. Another consequence of the theorem is the following particular case of Huppert's conjecture: Let G be a group of odd order. Assume that every normal Sylow subgroup of G is abelian. Then \(| \rho(G)| \leq 2\sigma(G)\), where \(\rho(G)=\cup_{\chi \in Irr(G)}\pi (\chi (1))\) and \(\sigma(G)=\max_{\chi \in Irr(G)}| \pi(\chi(1))|\).
scientific article; zbMATH DE number 4173220