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

TL;DR

This paper investigates how the ratio of degrees of nonlinear irreducible characters in a finite group constrains its nonabelian composition factors and the structure of the group.

Contribution

It establishes bounds on the size and number of nonabelian composition factors based on the character degree ratio, especially excluding groups isomorphic to PSL_2(q).

Findings

Bounds on the order and multiplicity of certain composition factors in terms of character degree ratio.

When PSL_2(q) groups are not factors, the index of the solvable radical is bounded by a power of the ratio.

Provides structural control of finite groups via character degree ratios.

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 $G$.