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

TL;DR

This paper applies recent theoretical results to verify Brauer's $k(B)$-Conjecture for blocks with certain defect groups, showing the number of irreducible characters is bounded by the defect group size under specific conditions.

Contribution

It extends the verification of Brauer's $k(B)$-Conjecture to blocks with minimal non-abelian defect groups and those with abelian defect groups lacking large elementary abelian summands.

Findings

Verified Brauer's $k(B)$-Conjecture for blocks with minimal non-abelian defect groups.

Established bounds on the number of irreducible characters for blocks with abelian defect groups.

Extended previous results by applying recent theoretical developments.

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.