The Sylow Subgroups of a Finite Reductive Group

TL;DR

This paper characterizes the structure of Sylow -subgroups in finite reductive groups, linking their properties to complex reflection groups and cyclotomic factors, extending to cases involving isogenies and Frobenius morphisms.

Contribution

It provides a detailed description of Sylow -subgroups in finite reductive groups, connecting their structure to complex reflection groups and cyclotomic factors, including cases with isogenies.

Findings

Sylow -subgroups governed by complex reflection groups

Structure depends on cyclotomic factors of the generic order

Extension to cases with isogenies and Frobenius morphisms

Abstract

We describe the structure of Sylow {\ell}-subgroups of a finite reduc-tive group G(Fq) when q $\not\equiv$ 0 (mod {\ell}) that we find governed by a complex reflection group attached to G and {\ell}, which depends on {\ell} only through the set of cyclotomic factors of the generic order of G(Fq) whose value at q is divisible by {\ell}. We also tackle the more general case G F where F is an isogeny whose a power is a Frobenius morphism.