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

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 $G$.Comment: 16 pages, 1 tabl