Cartan matrices and Brauer's k(B)-conjecture

Abstract: We show (among other things) that Brauer's k(B)-conjecture holds for defect groups with are central extensions of metacyclic 2-groups by cyclic groups. The same holds for defect groups which contain a central cyclic subgroup of index at most 9. In particular the k(B)-conjecture holds for 2-blocks of defect at most 4 and 3-blocks of defect at most 3. As a byproduct we obtain the block invariants of 2-blocks with minimal nonmetacyclic defect groups.