Some Properties of Twisted Chevalley Groups

TL;DR

This thesis explores the structural properties of twisted Chevalley groups over rings, focusing on normality, subgroup classification, and normalizers, providing new insights into their algebraic structure and subgroup relations.

Contribution

It offers new results on the normality of elementary subgroups, classifies subgroups normalized by elementary groups, and characterizes normalizers within ring extensions.

Findings

Normality of relative elementary subgroups established.

Classification of subgroups normalized by elementary groups provided.

Normalizers in ring extensions are shown to coincide and equal the original group for adjoint types.

Abstract

This thesis investigates certain structural properties of twisted Chevalley groups over commutative rings, focusing on three key problems. Let $R$ be a commutative ring satisfying mild conditions. Let $G_{\pi,\sigma} (\Phi, R)$ denote a twisted Chevalley group over $R$, and let $E'_{\pi, \sigma} (\Phi, R)$ denote its elementary subgroup. The first problem concerns the normality of $E'_{\pi, \sigma} (\Phi, R, J)$, the relative elementary subgroups at level $J$, in the group $G_{\pi, \sigma} (\Phi, R)$. The second problem addresses the classification of the subgroups of $G_{\pi, \sigma}(\Phi, R)$ that are normalized by $E'_{\pi, \sigma}(\Phi, R)$. This classification provides a comprehensive characterization of the normal subgroups of $E'_{\pi, \sigma}(\Phi, R)$. Lastly, the third problem investigates the normalizers of $E'_{\pi, \sigma}(\Phi, R)$ and $G_{\pi, \sigma}(\Phi, R)$ in the bigger group $G_{\pi, \sigma}(\Phi, S)$, where $S$ is a ring extension of $R$. We prove that these normalizers coincide. Moreover, for groups of adjoint type, we show that they are precisely equal to $G_{\pi, \sigma}(\Phi, R)$.