Alperin's bound and normal Sylow subgroups

TL;DR

This paper investigates a conjecture relating the number of p-Brauer characters to the normality of Sylow p-subgroups, proving it for p=2 and advancing understanding of Alperin's bounds in modular representation theory.

Contribution

It reduces a conjecture about p-Brauer characters to a simple group question and proves it for p=2, also improving bounds for 2-blocks of maximal defect.

Findings

Confirmed the conjecture for p=2.

Established a reduction theorem for Alperin's lower bound.

Improved bounds for 2-blocks of maximal defect.

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.