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

Abstract

Abstract. 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 E p (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 F 2 . 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 An important pendant to Theorem 1 is the following result, in which we explicitly calculate the normalizer of EEO(n, R, A). Namely, consider the reduction homomorphism ρ A : GL(n, R) −→ GL(n, R/A) and denote by CGO(n, R, A) the full preimage of the group GO(n, R) with respect to ρ A . The condition for a matrix to belong to CGO(n, R, A) is described by obvious quadratic congruences on its entries. Now we are in a position to state the second major result of the present paper. EEO(n, R, A) ≤ H ≤ N G (EEO(n, R, A)). Theorem 2. Under the assumptions of Theorem 1, for any ideal A R we have N G (EEO(n,...

Bak's work on theK-theory of rings

“…Also, the recent classification of subgroups of the general linear group GL.2n;R/ containing the elementary quadratic subgroup EU.2n;R;ƒ/ was made possible by employing the powerful form of Bak's "conjugation calculus" [114,169,170].…”

Overgroups of classical groups in linear group over Banach algebras

“…By the proof of Lemma 2.10, X contains an element η with the form I 2n + r(a ij ) 2n×2n , where r is a real number with sufficiently small |r|, which is not in GU 2n R. Proof. Since η = I 2n + r(a ij ) 2n×2n is not in GU 2n R, there exists ρ ij (a), without loss of generality, assume that ρ ij (a) = ρ 12 (1), such that ξ = ηρ 12 …”

“…In the resent years, Vavilov and Petrov [12,13], and the author [14] described the overgroups of symplectic and orthogonal groups (with hyperbolic form) over commutative rings; Petrov [6] also classified under a local stable rank condition with form parameter, the overgroups of unitary groups (with hyperbolic form).…”

Overgroups of the elementary unitary group in linear group over commutative rings

“…Vaserstein in [10][11][12][13] obtained some results on the general linear groups. The results on the overgroups of symplectic and orthogonal groups (with hyperbolic form) over commutative rings were given in [15][16][17]. Petrov in [9] investigated the overgroups of unitary groups (with hyperbolic form) under a local stable rank condition with form parameter.…”