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 $\hat{I}\mu \in \{\pm \}$, we define ${\mathrm{cd}}_{\hat{I}}\mu (G)=\{\ddot{I}{‡}_{\hat{I}}\mu (1)\mid \ddot{I}‡\in \mathrm{Irr}(G)\}$, where $\ddot{I}{‡}_{+}(1)=\ddot{I}‡(1)$ denotes the degree of $\ddot{I}‡$, $\ddot{I}{‡}_{-}(1)=|G:\ker (\ddot{I}‡)|/\ddot{I}‡(1)$ denotes the codegree of $\ddot{I}‡$. Further, let $\ddot{I}{‰}_{\hat{I}}\mu (G)=\{\ddot{I}€(n)\mid n\in {\mathrm{cd}}_{\hat{I}}\mu (G)\}$, where $\ddot{I}€(n)$ stands for the set of prime divisors of $n$. We established that if $|\ddot{I}{‰}_{\hat{I}}\mu (G)|\le 3$, then $G$ is solvable. Additionally, a generalization of this result is obtained in the case when $\hat{I}\mu =+$.