Some Properties of Twisted Chevalley Groups

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 $G_{\pi, \sigma}(\Phi, R)$.