Alperin's bound and normal Sylow subgroups

Let be a finite group, a prime number and a Sylow ‐subgroup of . Recently, Malle, Navarro, and Tiep conjectured that the number of ‐Brauer characters of coincides with that of the normalizer if and only if is normal in . We reduce this conjecture to a question about finite simple groups and prove it for the prime . As a by‐product of our work, we prove a reduction theorem for the blockwise version of Alperin's lower bound on ‐Brauer characters and prove it for 2‐blocks of maximal defect. This improves recent results obtained by Malle, Navarro, and Tiep.