Character degrees and local subgroups revisited

Abstract: Let $p$ and $q$ be different primes and let $G$ be a finite $q$-solvable group. We prove that $\mathrm{Irr}{p'}(G)\subseteq \mathrm{Irr}{q'}(G)$ if and only if $\mathbf{N}G(P)\subseteq \mathbf{N}_G(Q)$ and $\mathbf{C}{Q'}(P)=1$ for some $P\in\mathrm{Syl}_p(G)$ and $Q\in\mathrm{Syl}_q(G)$. Further, if $B$ is a $q$-block of $G$ and $p$ does not divide the degree of any character in $\mathrm{Irr}(B)$ then we prove that a Sylow $p$-subgroup of $G$ is normalized by a defect group of $B$. This removes the $p$-solvability condition of two theorems of G. Navarro and T. R. Wolf.