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arXiv · 2310.00134

Brauer's problem 21 for principal blocks

Alexander Moretó·Noelia Rizo·A. A. Schaeffer Fry
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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.

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Alexander Moretó, Noelia Rizo, A. A. Schaeffer Fry. 2023-09-29. Brauer's problem 21 for principal blocks. https://arxiv.org/abs/2310.00134

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