Height Zero Conjecture with Galois Automorphisms

We prove a strengthening of Brauer’s height zero conjecture for principal 2-blocks with Galois automorphisms. This requires a new extension of the Itô–Michler theorem for the prime~2, again with Galois automorphisms. We close, this time for odd primes $p$, with a new characterisation of $p$-closed groups via the decomposition numbers of certain characters.