Subgroups of Split Orthogonal Groups over a Commutative Ring

Abstract

The subgroups of the split orthogonal group Γ = GO(n,R) of degree n over a commutative ring R with 2 ∈R* that contain the elementary subgroup of a regularly embedded semisimple group F are described. It is shown that if the ranks of all irreducible components of F are at least 4, then the description of its overgroups is standard in the sense that for any intermediate subgroup H there exists a unique orthogonal net of ideals such that H lies between the corresponding net subgroup and its normalizer in Γ. A similar result for subgroups of the general linear group GL(n,R) with irreducible components of ranks at least 2 was obtained by Z. I. Borewicz and the present author. Examples which show that if F has irreducible components of ranks 2 or 3, then the standard description does not hold were constructed. The paper is based on the previous publications by the author, where similar results were obtained in some special cases, but the proof is based on a new computational trick. Bibliography: 76 titles.