Normalizer of Twisted Chevalley Groups over Commutative Rings

Abstract

Let $R$ be a commutative ring with unity. Consider the twisted Chevalley group $G_{\ddot{I}€,\ddot{I}ƒ}(\hat{I}¦,R)$ of type $\ddot{I}•$ over $R$ and its elementary subgroup $E_{\ddot{I}€,\ddot{I}ƒ}^{\prime }(\hat{I}¦,R)$. This paper 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 larger group $G_{\ddot{I}€,\ddot{I}ƒ}(\hat{I}¦,S)$, where $S$ is an extension ring of $R$. We establish that under certain conditions on $R$ these normalizers coincide. Moreover, in the case of adjoint type groups, we show that they are precisely equal to $G_{\ddot{I}€,\ddot{I}ƒ}(\hat{I}¦,R)$.

12 pages