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Problem 21 of Brauer's list of problems from 1963 asks whether for anypositive integer k there are finitely many isomorphism classes of groups thatoccur as the defect group of a block with k irreducible characters. We solvethis problem for principal blocks. Another long-standing open problem (from1982) in this area asks whether the defect group of a block with 3 irreduciblecharacters is necessarily the cyclic group of order 3. In most cases we reducethis problem to a question on simple groups that is closely related to therecent solution of Brauer's height zero conjecture.

Brauer's problem 21 for principal blocks