Brauer's problem 21 for principal blocks

TL;DR

This paper addresses longstanding open problems in modular representation theory by proving finiteness results for principal blocks and reducing complex conjectures to simpler cases involving simple groups.

Contribution

It solves Brauer's Problem 21 for principal blocks and connects the defect group conjecture with recent developments in the height zero conjecture.

Findings

Finiteness of isomorphism classes of groups as defect groups for principal blocks.

Reduction of the defect group conjecture to simple groups.

Connection to the recent solution of Brauer's height zero conjecture.

Abstract

Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group of a block with k irreducible characters. We solve this problem for principal blocks. Another long-standing open problem (from 1982) in this area asks whether the defect group of a block with 3 irreducible characters is necessarily the cyclic group of order 3. In most cases we reduce this problem to a question on simple groups that is closely related to the recent solution of Brauer's height zero conjecture.