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 $ε\in \{ \pm \}$, we define $\mathrm{cd}_ε(G)=\{ χ_ε(1)\mid χ\in \mathrm{Irr}(G) \}$, where $χ_{+}(1)=χ(1)$ denotes the degree of $χ$, $χ_{-}(1)=|G:\ker(χ)|/χ(1)$ denotes the codegree of $χ$. Further, let $ω_ε(G)=\{ π(n)\mid n\in \mathrm{cd}_ε(G) \}$, where $π(n)$ stands for the set of prime divisors of $n$. We established that if $|ω_ε(G)|\leq 3$, then $G$ is solvable. Additionally, a generalization of this result is obtained in the case when $ε=+$.