On Normal Subgroups of Twisted Chevalley Groups over Commutative Rings

TL;DR

This paper establishes structure theorems for twisted Chevalley groups over commutative rings, focusing on normal subgroups and subgroup classification, with specific results for semilocal rings.

Contribution

It proves the normality of elementary congruence subgroups and classifies subgroups normalized by the elementary subgroup in twisted Chevalley groups.

Findings

Normality of elementary congruence subgroups established

Classification of subgroups normalized by elementary subgroup

Decomposition of groups over semilocal rings into elementary and torus parts

Abstract

In this paper, we prove two structure theorems for twisted Chevalley groups $G_\sigma (R)$ over a commutative ring $R$ with unity. The first theorem concerns the normality of $E'_\sigma (R,J)$, the elementary congruence subgroups at level $J$, in the group $G_\sigma (R)$. The second theorem classifies all subgroups of $G_\sigma (R)$ normalized by its elementary subgroup $E'_\sigma (R)$. Along the way, we obtain several interesting results. For instance, when $R$ is a semilocal ring, we show that $G_\sigma(R)$ can be expressed as the (internal) product of $E'_\sigma (R)$ and the maximal torus $T_\sigma (R)$ of $G_\sigma (R)$.