On Normal Subgroups of Twisted Chevalley Groups over Commutative Rings

Abstract

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

The appendix to this paper was contributed by Pavel Gvozdevsky