Controlling composition factors of a finite group by its character degree ratio

Abstract

For a finite nonabelian group $G$ let $\rat(G)$ be the largest ratio of degrees of two nonlinear irreducible characters of $G$. We show that nonabelian composition factors of $G$ are controlled by $\rat(G)$ in some sense. Specifically, if $S$ different from the simple linear groups $\PSL_2(q)$ is a nonabelian composition factor of $G$, then the order of $S$ and the number of composition factors of $G$ isomorphic to $S$ are both bounded in terms of $\rat(G)$. Furthermore, when the groups $\PSL_2(q)$ are not composition factors of $G$, we prove that $|G:\Oinfty(G)|\leq \rat(G)^{21}$ where $\Oinfty(G)$ denotes the solvable radical of $G$.