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A Brauer--Galois height zero conjecture

Gunter Malle, Alexander Moretó, Noelia Rizo, A. A. Schaeffer Fry

Abstract

Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any prime $p$ we consider a certain elementary abelian $p$-subgroup of the absolute Galois group and propose a Galois version of Brauer's height zero conjecture for principal $p$-blocks. We prove it when $p=2$ and also for arbitrary $p$ when $G$ does not involve certain groups of Lie type of small rank as composition factors. Furthermore, we prove it for almost simple groups and for $p$-solvable groups.

A Brauer--Galois height zero conjecture

Abstract

Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal -blocks when , considering a particular Galois automorphism of order~. In this paper, for any prime we consider a certain elementary abelian -subgroup of the absolute Galois group and propose a Galois version of Brauer's height zero conjecture for principal -blocks. We prove it when and also for arbitrary when does not involve certain groups of Lie type of small rank as composition factors. Furthermore, we prove it for almost simple groups and for -solvable groups.
Paper Structure (8 sections, 27 theorems, 12 equations)

This paper contains 8 sections, 27 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Almost simple and quasi-simple groups
  3. Extensions of Itô--Michler for almost simple groups
  4. Some observations on extending unipotent characters
  5. Existence of certain character extensions
  6. Extension of Brauer's height zero conjecture for almost simple groups
  7. The Galois Itô--Michler theorem
  8. The Galois height zero conjecture

Key Result

Theorem 2

Let $G$ be a finite group, let $p$ be a prime number and let $P\in{\operatorname{Syl}}_p(G)$. Suppose that $G$ does not have composition factors isomorphic to $S$ with $(S,p)$ a pair as listed in Theorem thm:as. Then Conjecture galcon holds for $G$.

Theorems & Definitions (55)

  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 45 more