Height zero characters and Galois automorphisms

TL;DR

This work strengthens Brauer's height zero conjecture for the principal $p$-block by incorporating a Galois-action framework via $\operatorname{Irr}_{\mathcal{J}}(B_0(G))$, resolving a conjecture of MMRSF and providing a Galois analogue of Itô–Michler. The authors replace the infinite Galois object with a finite model $\Omega(|G|)$ to facilitate CFSG-based classifications and MinimalHeights techniques, enabling precise control over normal structure and character degrees. The main results are Theorem A, which proves that $P\in\mathrm{Syl}_p(G)$ is abelian whenever all $\mathcal{J}$-invariant irreducibles in the principal block have $p'$-degree, and Theorem B, which gives a sharp decomposition ${\mathbf O}^{p'}(G)={\mathbf O}_p(G)\times K$ with $K/{\mathbf O}_{p'}(K)$ a direct product of nonabelian simple groups of order divisible by $p$ lacking $\Omega$-invariant $p$-degree characters. Collectively, these results constitute a strong Galois-generalization of Itô–Michler and yield structural constraints on finite groups under Galois actions, with techniques anchored in finite-model reductions and deep permutation-group classifications.

Abstract

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.

Theorems & Definitions (31)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof