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 $\chi \in {\rm Irr}(G)$, we write ${\rm cod}(\chi)=|G:{\rm ker} \chi|/\chi(1)$, and called it the codegree of the irreducible character $\chi$. Let $N\unlhd G$, write ${\rm Irr}(G|N)={\chi \in {\rm Irr}(G)~|~N\nsubseteq {\rm ker}\chi}$, and ${\rm cod}(G|N)={ {\rm cod}(\chi) ~|~\chi\in{\rm Irr}(G|N)}.$ In this Ipaper, we prove that if $N\unlhd G$ and every member of ${\rm cod}(G|N')$ is not divisible by some fixed prime $p\in \pi(G)$, then $N$ has a normal $p$-complement and $N$ is solvable.