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

In this note we give applications of recent results coming mostly from the third paper of this series. It is shown that the number of irreducible characters in a p-block of a finite group with abelian defect group D is bounded by vertical bar D vertical bar (Brauer's k(B)-conjecture) provided D has no large elementary abelian direct summands. Moreover, we verify Brauer's k(B)conjecture for all blocks with minimal non-abelian defect groups. This extends previous results by various authors.