On Normal Subgroups of Twisted Chevalley Groups over Commutative Rings

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