Alperin-McKay implies Brauer's problem 21 (Q1911645)

A connection between two problems in modular representation theory of finite groups is proved, based on Zelmanov's solution of the restricted Burnside problem. R. Brauer's Problem 21 is whether the order of the defect group of a \(p\)-block is bounded in terms of the number of ordinary irreducible characters in the block. The Alperin-McKay conjecture asserts that a \(p\)-block and its Brauer correspondent have the same number of characters of height 0.