Cartan matrices and Brauerʼs k(B)-conjecture II

This paper continues Sambale (2011) [28], . We show that the methods developed there also work for odd primes. In particular we prove Brauerʼs k(B)-conjecture for defect groups which contain a central, cyclic subgroup of index at most 9. As a consequence, the k(B)-conjecture holds for 3-blocks of defect at most 3. In the second part of the paper we illustrate the limits of our methods by considering an example. Then we use the work of Kessar, Koshitani and Linckelmann [13], (and thus the classification) to show that the k(B)-conjecture is satisfied for 2-blocks of defect 5 except for the extraspecial defect group D8⁎D8. As a byproduct we also obtain the block invariants of 2-blocks with minimal nonmetacyclic defect groups. Some proofs rely on computer computations with GAP (The GAP Group, 2008 [10]).