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 $G_{\ddot{I}€,\ddot{I}ƒ}(\hat{I}¦,R)$ denote a twisted Chevalley group over $R$, and let $E_{\ddot{I}€,\ddot{I}ƒ}^{\prime }(\hat{I}¦,R)$ denote its elementary subgroup. The first problem concerns the normality of $E_{\ddot{I}€,\ddot{I}ƒ}^{\prime }(\hat{I}¦,R,J)$, the relative elementary subgroups at level $J$, in the group $G_{\ddot{I}€,\ddot{I}ƒ}(\hat{I}¦,R)$. The second problem addresses the classification of the subgroups of $G_{\ddot{I}€,\ddot{I}ƒ}(\hat{I}¦,R)$ that are normalized by $E_{\ddot{I}€,\ddot{I}ƒ}^{\prime }(\hat{I}¦,R)$. This classification provides a comprehensive characterization of the normal subgroups of $E_{\ddot{I}€,\ddot{I}ƒ}^{\prime }(\hat{I}¦,R)$. Lastly, the third problem investigates the normalizers of $E_{\ddot{I}€,\ddot{I}ƒ}^{\prime }(\hat{I}¦,R)$ and $G_{\ddot{I}€,\ddot{I}ƒ}(\hat{I}¦,R)$ in the bigger group $G_{\ddot{I}€,\ddot{I}ƒ}(\hat{I}¦,S)$, 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 $G_{\ddot{I}€,\ddot{I}ƒ}(\hat{I}¦,R)$.

PhD thesis, IIT Bombay, 2025