Principal 2-Blocks and Sylow 2-Subgroups

Let 퐺 be a finite group with Sylow 2‐subgroup 푃⩽퐺 . Navarro–Tiep–Vallejo have conjectured that the principal 2‐block of 푁퐺(푃) contains exactly one irreducible Brauer character if and only if all odd‐degree ordinary irreducible characters in the principal 2‐block of 퐺 are fixed by a certain Galois automorphism 휎∈Gal(ℚab/ℚ) . Recent work of Navarro–Vallejo has reduced this conjecture to almost simple groups. We show that the conjecture holds for all relevant almost simple groups, thus establishing the conjecture for all finite groups.