Characters of relative p'-degree over normal subgroups

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