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.
Benjamin SAMBALE. "Cartan matrices and Brauer's $k(B)$-conjecture IV." J. Math. Soc. Japan 69 (2) 735 - 754, April, 2017. https://doi.org/10.2969/jmsj/06920735