Characters of relative p′-degree over normal subgroups

Let Z be a normal subgroup of a finite group G, let lambda is an element of 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 is an element of 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.