We prove that Brauer’s
height zero conjecture holds for p-blocks of finite groups with metacyclic defect
groups. If the defect group is nonabelian and contains a cyclic maximal subgroup, we
obtain the distribution into p-conjugate and p-rational irreducible characters. The
Alperin–McKay conjecture then follows provided p = 3. Along the way we verify a
few other conjectures. Finally we consider more closely the extraspecial defect group
of order p3 and exponent p2 for an odd prime. Here for blocks with inertial
index 2 we prove the Galois–Alperin–McKay conjecture by computing k0(B).
Then for p ≤ 11 also Alperin’s weight conjecture follows. This improves
results of Gao (2012), Holloway, Koshitani, Kunugi (2010) and Hendren
(2005).
Keywords
Brauer’s height zero conjecture, metacyclic defect groups,
Alperin’s weight conjecture
