## Journal of the Mathematical Society of Japan

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

Benjamin SAMBALE

#### 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 69, Number 2 (2017), 735-754.

Dates
First available in Project Euclid: 20 April 2017

https://projecteuclid.org/euclid.jmsj/1492653645

Digital Object Identifier
doi:10.2969/jmsj/06920735

Mathematical Reviews number (MathSciNet)
MR3638283

Zentralblatt MATH identifier
06737032

Subjects
Primary: 20C15: Ordinary representations and characters
Secondary: 20C20: Modular representations and characters

#### Citation

SAMBALE, Benjamin. Cartan matrices and Brauer's $k(B)$-conjecture IV. J. Math. Soc. Japan 69 (2017), no. 2, 735--754. doi:10.2969/jmsj/06920735. https://projecteuclid.org/euclid.jmsj/1492653645

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