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 ${\mathrm{Irr}}_{p^{\prime }}(G)\subseteq {\mathrm{Irr}}_{q^{\prime }}(G)$ if and only if ${\mathbf{N}}_G(P)\subseteq {\mathbf{N}}_G(Q)$ and ${\mathbf{C}}_{Q^{\prime }}(P)=1$ 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.

7 pages; Final version to appear in J. Algebra