We prove that the number of irreducible ordinary characters in the principal $p$-block of a finite group $G$ of order divisible by $p$ is always at least $2\sqrt{p-1}$. This confirms a conjecture of Héthelyi and Külshammer (2000) for principal blocks and provides an affirmative answer to Brauer’s problem 21 (1963) for principal blocks of bounded defect. Our proof relies on recent works of Maróti (2016) and Malle and Maróti (2016) on bounding the conjugacy class number and the number of $p^{\prime }$-degree irreducible characters of finite groups, earlier works of Broué, Malle and Michel (1993) and Cabanes and Enguehard (2004) on the distribution of characters into unipotent blocks and $e$-Harish-Chandra series of finite reductive groups, and known cases of the Alperin–McKay conjecture.

我们证明了主体中不可约普通字符的数量$p$-有限群的块$G$可被整除的顺序$p$总是至少$\mathrm{2个}\sqrt{p-\mathrm{1个}}$. 这证实了 Héthelyi 和 Külshammer (2000) 对主块的猜想，并为布劳尔问题 21 (1963) 的有界缺陷主块提供了肯定的答案。我们的证明依赖于 Maróti (2016) 和 Malle 和 Maróti (2016) 关于限制共轭类数和$p^{‘}$-degree irreducible characters of finite groups，Broué、Malle 和 Michel（1993）以及 Cabanes 和 Enguehard（2004）关于将特征分布到单能块中的早期著作和$\mathrm{电子}$-Harish-Chandra 系列的有限还原群，以及 Alperin-McKay 猜想的已知案例。