Character degrees and local subgroups of $\pi$-separable groups

Abstract:

Let $G$ be a finite $\{p,q \}$-solvable group for different primes $p$ and $q$. Let $P \in \text {Syl}_{p}(G)$ and $Q \in \text {Syl}_{q}(G)$ be such that $PQ=QP$. We prove that every $\chi \in \text {Irr}(G)$ of $p^{\prime }$-degree has $q^{\prime }$-degree if and only if $\mathbf {N}_{G}(P) \subseteq \mathbf {N}_{G}(Q)$ and $\mathbf {C}_{Q^{\prime }}(P)=1$.