  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 $G$.
