$M$-groups and Codegrees; $M_p$-groups and Brauer Character Degrees

Let $G$ be a finite group and $p$ be a prime. We prove that if $G$ has three codegrees, then $G$ is an $M$-group. We prove for some prime $p$ that if every irreducible Brauer character of $G$ is a prime, then for every normal subgroup $N$ of $G$ either $G\mathrm{/}N$ or $N$ is an $M_p$-group.