Normal $p$-complements and irreducible character codegrees

Abstract

Let $G$ be a finite group and $p\in \pi (G)$, and let Irr$(G)$ be the set of all irreducible complex characters of $G$. Let $\chi \in \mathrm{Irr}(G)$, we write $\mathrm{cod}(\chi )=|G:\ker \chi |/\chi (1)$, and called it the codegree of the irreducible character $\chi$. Let $N⊴G$, write $\mathrm{Irr}(G|N)=\{\chi \in \mathrm{Irr}(G)|N⊈\ker \chi \}$, and $\mathrm{cod}(G|N)=\{\mathrm{cod}(\chi )|\chi \in \mathrm{Irr}(G|N)\}.$ In this Ipaper, we prove that if $N⊴G$ and every member of $\mathrm{cod}(G|N^{\prime })$ is not divisible by some fixed prime $p\in \pi (G)$, then $N$ has a normal $p$-complement and $N$ is solvable.

The result has been proved by other authors