Galois Automorphisms on Harish-Chandra Series and Navarro's Self-Normalizing Sylow $2$-Subgroup Conjecture

TL;DR

This paper proves Navarro's conjecture on self-normalizing Sylow 2-subgroups by analyzing Galois automorphisms on Harish-Chandra series and verifying conditions for simple groups.

Contribution

It completes the proof of Navarro's conjecture by describing Galois automorphisms' action on Harish-Chandra characters and confirming the conditions for all simple groups.

Findings

Navarro's conjecture is confirmed for all simple groups.

Galois automorphisms act compatibly with Harish-Chandra parametrization.

The proof reduces the conjecture to properties of simple groups.

Abstract

G. Navarro has conjectured a necessary and sufficient condition for a finite group $G$ to have a self-normalizing Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of $G$. In a previous article, the author has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the required conditions.