Height Zero Conjecture with Galois Automorphisms

TL;DR

This paper advances the understanding of Brauer's height zero conjecture by proving a strengthened version for principal 2-blocks with Galois automorphisms, extending key theorems and characterizing p-closed groups.

Contribution

It introduces new extensions of the Itô–Michler theorem for prime 2 and characterizes p-closed groups through decomposition numbers, with implications for Galois automorphisms.

Findings

Proved a strengthened height zero conjecture for principal 2-blocks with Galois automorphisms.

Extended the Itô–Michler theorem to include Galois automorphisms for prime 2.

Provided a new characterization of p-closed groups via character decomposition numbers.

Abstract

We prove a strengthening of Brauer's height zero conjecture for principal 2-blocks with Galois automorphisms. This requires a new extension of the It\^o--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.