2-Blocks with minimal nonabelian defect groups

TL;DR

This paper investigates numerical invariants of 2-blocks with minimal nonabelian defect groups, proving several conjectures and expanding understanding of their structure, especially for two key families of such groups.

Contribution

It proves Brauer's k(B)-conjecture and the Olsson-conjecture for all 2-blocks with minimal nonabelian defect groups, and verifies Alperin's weight and Dade's conjectures for one family.

Findings

Proved Brauer's k(B)-conjecture for these blocks.

Confirmed Olsson's conjecture for all cases.

Validated Alperin's weight and Dade's conjectures for one family.

Abstract

We study numerical invariants of 2-blocks with minimal nonabelian defect groups. These groups were classified by R\'edei. If the defect group is also metacyclic, then the block invariants are known. In the remaining cases there are only two (infinite) families of "interesting" defect groups. In all other cases the blocks are nilpotent. We prove Brauer's k(B)-conjecture and the Olsson-conjecture for all 2-blocks with minimal nonabelian defect groups. For one of the two families we also show that Alperin's weight conjecture and Dade's conjecture is satisfied. This paper is a part of the author's PhD thesis.