Normalizers of Sylow subgroups in finite reflection groups

TL;DR

This paper investigates the structure of normalizers of Sylow subgroups within finite reflection groups, providing a decomposition framework that extends understanding of subgroup symmetries in both real and complex cases.

Contribution

It establishes a semidirect product decomposition of Sylow subgroup normalizers using minimal parabolic subgroups and known decompositions, generalizing previous results.

Findings

Decomposition of normalizers in terms of parabolic subgroups

Existence of Sylow $\ell$-subgroups stable under diagram automorphisms

Applicable to both real and complex reflection groups

Abstract

Let $W$ be a finite reflection group, either real or complex, and $S_\ell$ a Sylow $\ell$-subgroup of $W$. We prove the existence of a semidirect product decomposition of $N_W(S_\ell)$ in terms of the unique parabolic subgroup of $W$ minimally containing $S_\ell$ and known decompositions of normalizers of parabolic subgroups. In the real setting, the description follows from the existence of Sylow $\ell$-subgroups stable under the Coxeter diagram automorphisms of finite reflection groups with no proper parabolic subgroup containing a Sylow $\ell$-subgroup.