Alperin's bound and normal Sylow subgroups

TL;DR

This work reduces the Malle–Navarro–Tiep conjecture to questions about finite simple groups via an inductive Alperin bound framework, and proves the conjecture for the prime $p=2$ by verifying a blockwise inductive Alperin bound for all blocks of maximal defect of covering groups of non-abelian simple groups. A blockwise reduction theorem links the inductive bound for simple groups to the general case, while a block-free version via Sylow-AWC is also established. The authors give a detailed analysis of simple groups, showing no simple group can be a counterexample, and provide a complete verification for $2$-blocks of maximal defect across quasi-simple and covering groups, including groups of Lie type and alternating groups. Consequently, the MNT conjecture holds for all finite groups at $p=2$, and a blockwise lower bound for Alperin's weight is obtained in the maximal-defect setting, refining recent results in the area.

Abstract

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.

Theorems & Definitions (55)

• Conjecture 1.1: Malle--Navarro--Tiep
• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Conjecture 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Lemma 2.5