Normal $p$-complements and irreducible character codegrees

TL;DR

This paper proves that if all irreducible character codegrees associated with a normal subgroup are not divisible by a fixed prime, then the subgroup has a normal p-complement and is solvable.

Contribution

It establishes a new criterion linking character codegrees to the existence of normal p-complements and solvability of subgroups.

Findings

Normal p-complement exists under the given codegree condition.

Subgroup N is solvable if codegrees are not divisible by p.

Provides a character-theoretic criterion for subgroup structure.

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.