Solvable groups whose monomial, monolithic characters have prime power codegrees

Abstract

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

To appear in International Journal of Group Theory