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

Abstract

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 $|D|$ (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.