Alperin's bound and normal Sylow subgroups

2025/02/18 by Feng, Zhicheng, Martínez, J. Miquel, Rossi, Damiano
#20C15 #20C20 #20C33 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2502.12841

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.

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