标题： 主块的 Brauer 问题 21 显示英文标题

标题： Brauer's problem 21 for principal blocks

摘要： 布劳尔在1963年列出的问题清单中的第21个问题询问，对于任意正整数k，是否存在有限个同构类的群作为具有k个不可约特征标的块的亏群。 我们解决了主块的这个问题。 该领域另一个长期悬而未决的问题（来自1982年）询问具有3个不可约特征标的块的亏群是否必然是3阶循环群。 在大多数情况下，我们将此问题简化为一个与布劳尔高度零猜想近期解决方案密切相关的关于单群的问题。 显示英文摘要

摘要： 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.