The E-normal structure of odd dimensional unitary groups

Abstract

In this paper we define odd dimensional unitary groups $U_{2n+1}(R,\Delta )$. These groups contain as special cases the odd dimensional general linear groups $G{L}_{2n+1}(R)$ where $R$ is any ring, the odd dimensional orthogonal and symplectic groups $O_{2n+1}(R)$ and $S{p}_{2n+1}(R)$ where $R$ is any commutative ring and further the first author's even dimensional unitary groups $U_{2n}(R,\Lambda )$ where $(R,\Lambda )$ is any form ring. We classify the E-normal subgroups of the groups $U_{2n+1}(R,\Delta )$ (i.e. the subgroups which are normalized by the elementary subgroup $E{U}_{2n+1}(R,\Delta )$), under the condition that $R$ is either a semilocal or quasifinite ring with involution and $n\ge 3$. Further we investigate the action of $U_{2n+1}(R,\Delta )$ by conjugation on the set of all E-normal subgroups.