Characters of relative $p’$-degree over normal subgroups

Abstract

Let $Z$ be a normal subgroup of a finite group $G$, let $\lambda \in \mathrm{Irr}(Z)$ be an irreducible complex character of $Z$, and let $p$ be a prime number. If $p$ does not divide the integers $\chi(1)/\lambda(1)$ for all $\chi \in \mathrm{Irr}(G)$ lying over $\lambda$, then we prove that the Sylow $p$-subgroups of $G/Z$ are abelian. This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary finite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture.