Height zero characters and Galois automorphisms

Let G be a finite group and let p be a prime. In this paper, we prove a strengthened version of Brauer’s height zero conjecture for the principal p -block of G that takes the action of a certain group of Galois automorphisms into account. This answers a conjecture recently proposed by Malle, Moretó, Rizo and Schaeffer Fry. We then use this to obtain a structural result which can be seen as a Galois version of the Itô–Michler theorem.