Character degrees and local subgroups revisited

Let p and q be different primes and let G be a finite q-solvable group. We prove that ${\mathrm{Irr}}_{p^{\prime }}\left(G\right)\subseteq {\mathrm{Irr}}_{q^{\prime }}\left(G\right)$ if and only if $N_G\left(P\right)\subseteq N_G\left(Q\right)$ and $C_{Q^{\prime }}\left(P\right)=1$ for some $P\in {\mathrm{Syl}}_p\left(G\right)$ and $Q\in {\mathrm{Syl}}_q\left(G\right)$. Further, if B is a q-block of G and p does not divide the degree of any character in $\mathrm{Irr}\left(B\right)$ 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.