OVERGROUPS OF EO(n,R)

Let R be a commutative ring with 1, n a natural number, and let l = [n/2]. Suppose that 2 ∈ R∗ and l ≥ 3. We describe the subgroups of the general linear group GL(n,R) that contain the elementary orthogonal group EO(n,R). The main result of the paper says that, for every intermediate subgroup H, there exists a largest ideal A R such that EEO(n,R,A) = EO(n,R)E(n,R,A) H. Another important result is an explicit calculation of the normalizer of the group EEO(n,R,A). If R = K 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 EO(2l, R) and the elementary symplectic group Ep(2l, R), analogous results appeared in previous papers by the authors (Zapiski Nauchn. Semin. POMI, 2000, v. 272; Algebra i Analiz, 2003, v. 15, no. 3). The present work, together with [19], constitutes a final and fully developed version of the paper “Overgroups of classical groups over commutative rings”, which was predicted in §11 of the survey [92]. Together with our papers [18, 19] it completely describes the subgroups of the general linear group G = GL(n,R) over a commutative ring R, containing the elementary subgroup of a split classical group in the vector representation. In [78], the second-named author generalized these results to the even unitary groups. Namely, in [18] we described the overgroups of EO(2l, R) under the assumption that l ≥ 3, 2 ∈ R∗, and in [19] we described the overgroups of Ep(2l, R) under the assumption that either l ≥ 3, or l = 2 and the ring R does not have the residue field F2. In the present paper, we consider the last remaining case; namely, we describe the overgroups of EO(n,R), where n = 2l + 1. As in the papers [18, 19], it turns out that when n ≥ 7 and 2 ∈ R∗, for every such subgroup H there exists a unique ideal A in R such that H lies between the group EEO(n,R,A) = EO(n,R)E(n,R,A) and its normalizer in GL(n,R). Since the even-dimensional case is very similar to the odd-dimensional one, and is in fact slightly easier, in the present paper we also give another proof, and a refinement of the results of [18]. The principal result of the present paper can be stated as follows. Theorem 1. Let R be a commutative ring. Suppose that 2 ∈ R∗ and n ≥ 7. Then for every subgroup H in G = GL(n,R) containing the elementary orthogonal group EO(n,R) there exists a unique ideal A R such that EEO(n,R,A) ≤ H ≤ NG(EEO(n,R,A)). 2000 Mathematics Subject Classification. Primary 20G35.