On prime divisors of character degrees and codegrees

Abstract

Let $G$ be a finite group, and let $\mathrm{Irr}(G)$ denote the set of irreducible complex characters of $G$. For $\epsilon\in \{ \pm \}$, we define $\mathrm{cd}_{\epsilon}(G)=\{ \chi_{\epsilon}(1)\mid \chi\in \mathrm{Irr}(G) \}$, where $\chi_{+}(1)=\chi(1)$ denotes the degree of $\chi$, $\chi_{-}(1)=|G:\ker(\chi)|/\chi(1)$ denotes the codegree of $\chi$. Further, let $\omega_{\epsilon}(G)=\{ \pi(n)\mid n\in \mathrm{cd}_{\epsilon}(G) \}$, where $\pi(n)$ stands for the set of prime divisors of $n$. We established that if $|\omega_{\epsilon}(G)|\leq 3$, then $G$ is solvable. Additionally, a generalization of this result is obtained in the case when $\epsilon=+$.