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


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.


Gabriel Navarro

Departament d'Àlgebra, Universitat de València, València, Spain

Pham Huu Tiep

University of Arizona, Tucson, AZ