标题： 阿尔佩林的界和正规西洛子群 显示英文标题

标题： Alperin's bound and normal Sylow subgroups

摘要： 设$G$为一个有限群，$p$为一个素数，$P$为$G$的一个西罗$p$-子群。 Recently, G. Malle, G. Navarro, and P. H. Tiep conjectured that the number of $p$-Brauer characters of $G$ coincides with that of the normaliser ${\bf N}_G(P)$ if and only if $P$ is normal in $G$. We reduce this conjecture to a question about finite simple groups and prove it for the prime $p = 2$. 作为我们工作的副产品，我们证明了关于$p$-Brauer特征标Alperin下界的块版本的约化定理，并将其证明适用于$2$-块的最大缺陷。 这改进了Malle、Navarro和Tiep最近获得的结果。 显示英文摘要

摘要： Let $G$ be a finite group, $p$ a prime number and $P$ a Sylow $p$-subgroup of $G$. Recently, G. Malle, G. Navarro, and P. H. Tiep conjectured that the number of $p$-Brauer characters of $G$ coincides with that of the normaliser ${\bf N}_G(P)$ if and only if $P$ is normal in $G$. We reduce this conjecture to a question about finite simple groups and prove it for the prime $p = 2$. As a by-product of our work, we prove a reduction theorem for the blockwise version of Alperin's lower bound on $p$-Brauer characters and prove it for $2$-blocks of maximal defect. This improves recent results obtained by Malle, Navarro, and Tiep.