Brauer's problem 21 for principal blocks
Group Theory
2023-10-03 v1 Representation Theory
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.
Keywords
Cite
@article{arxiv.2310.00134,
title = {Brauer's problem 21 for principal blocks},
author = {Alexander Moretó and Noelia Rizo and A. A. Schaeffer Fry},
journal= {arXiv preprint arXiv:2310.00134},
year = {2023}
}