Principal $2$-Blocks and Sylow $2$-Subgroups

TL;DR

This paper proves Navarro-Tiep-Vallejo's conjecture relating principal 2-blocks and Galois automorphisms for all finite groups by confirming it for all finite simple groups.

Contribution

The paper confirms the Navarro-Tiep-Vallejo conjecture for all finite simple groups, thereby establishing its validity for all finite groups.

Findings

Conjecture holds for all finite simple groups.

Principal 2-blocks' properties relate to Galois automorphisms.

Reduction to simple groups is successful.

Abstract

Let $G$ be a finite group with Sylow $2$-subgroup $P \leqslant G$. Navarro-Tiep-Vallejo have conjectured that the principal $2$-block of $N_G(P)$ contains exactly one irreducible Brauer character if and only if all odd-degree ordinary irreducible characters in the principal $2$-block of $G$ are fixed by a certain Galois automorphism $\sigma \in \mathrm{Gal}(\mathbb{Q}_{|G|}/\mathbb{Q})$. Recent work of Navarro-Vallejo has reduced this conjecture to a problem about finite simple groups. We show that their conjecture holds for all finite simple groups, thus establishing the conjecture for all finite groups.