Overgroups of $\mathrm{EO}(n,R)$

Abstract

Let be a commutative ring with 1, a natural number, and let . Suppose that and . We describe the subgroups of the general linear group that contain the elementary orthogonal group . The main result of the paper says that, for every intermediate subgroup , there exists a largest ideal such that . Another important result is an explicit calculation of the normalizer of the group . If is a field, similar results were obtained earlier by Dye, King, Shang Zhi Li, and Bashkirov. For overgroups of the even split elementary orthogonal group and the elementary symplectic group , analogous results appeared in previous papers by the authors (Zapiski Nauchn. Semin. POMI, 2000, v. 272; Algebra i Analiz, 2003, v. 15, no. 3).