The Reductive Subgroups of G_2

TL;DR

This paper classifies all reductive subgroups of the algebraic group G_2 and quasi-simple subgroups of its finite form G_2(q) over algebraically closed fields of positive characteristic.

Contribution

It provides a complete classification of reductive subgroups of G_2 and quasi-simple subgroups of G_2(q) in the defining characteristic, filling a gap in the subgroup structure theory.

Findings

All reductive subgroups of G_2 are identified.

All quasi-simple subgroups of G_2(q) are classified.

The results hold over algebraically closed fields of characteristic p>0.

Abstract

Let $G:=G_2(K)$ be a simple algebraic group of type $G_2$ defined over an algebraically closed field $K$ of characteristic $p>0$. Let $\sigma$ denote a standard Frobenius automorphism of $G$ such that $G_\sigma\cong G_2(q)$ with $q\geq 4$. In this paper we find all reductive subgroups of $G$ and quasi-simple subgroups of $G_\sigma$ in the defining characteristic.