Height-zero characters and normal subgroups in p-solvable groups

Fix a prime number p, and let N be a normal subgroup of a finite p-solvable group G. Let b be a p-block of N and suppose B is a p-block of G covering b. Let D be a defect group for the Fong-Reynolds correspondent of B with respect to b and let Bˆ be the unique p-block of NNG(D) having defect group D and inducing B. Suppose, further, that μ∈Irr(b), and let Irr0(B|μ) be the set of irreducible characters in B of height zero that lie over μ. We show that the number of characters in Irr0(B|μ) is equal to the number of characters in ⋃tIrr0(Bˆ|μt), where t runs through the inertial group T of b in G. This result generalizes a theorem of T. Okuyama and M. Wajima, which confirms the Alperin-McKay conjecture for p-solvable groups.