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Normal subgroups of orthogonal groups over commutative rings. (English) Zbl 0654.20052

Let A be a commutative ring with identity, V a right A-module, and q a quadratic form on V. The aim of the present paper is to give a description of the normal subgroups of the orthogonal group O(q,A). To do this, the author introduces a subgroup E(q,A) and an overgroup G(q,A) of O(q,A), and, under reasonable mild assumptions on the quadratic form q, uses localization and patching to derive a complete description of all subgroups of G(q,A) which are normalized by E(q,A). This is the best result up to now. The result was announced by the author at the Sino-US Seminar on Classical Groups and Related Areas, Beijing, May, 18-23, 1987.
Reviewer: Li Fuan

MSC:

20G35 Linear algebraic groups over adèles and other rings and schemes
20E07 Subgroup theorems; subgroup growth
11E16 General binary quadratic forms
20H25 Other matrix groups over rings
20E15 Chains and lattices of subgroups, subnormal subgroups
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