Solvable groups whose monomial, monolithic characters have prime power codegrees

TL;DR

This paper investigates the structure of solvable groups with specific character properties, proving that certain Sylow subgroups are normal if all monomial, monolithic characters have prime power codegrees.

Contribution

It establishes a new criterion linking prime power codegrees of monomial, monolithic characters to the normality of Sylow subgroups in solvable groups.

Findings

Sylow p-subgroups are normal under the given conditions

Prime power codegrees characterize certain group normality properties

Results apply to both ordinary and modular characters

Abstract

In this note, we prove that if $G$ is solvable and ${\rm cod}(\chi)$ is a $p$-power for every nonlinear, monomial, monolithic $\chi\in {\rm Irr}(G)$ or every nonlinear, monomial, monolithic $\chi \in {\rm IBr} (G)$, then $P$ is normal in $G$, where $p$ is a prime and $P$ is a Sylow $p$-subgroup of $G$.