TY - JOUR

T1 - Overgroups of EO(n, R)

AU - Vavilov, N. A.

AU - Petrov, V. A.

PY - 2008

Y1 - 2008

N2 - 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).

AB - 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).

KW - General linear group

KW - Overgroup

KW - Split elementary orthogonal group

UR - http://www.scopus.com/inward/record.url?scp=85009738238&partnerID=8YFLogxK

U2 - 10.1090/S1061-0022-08-00992-8

DO - 10.1090/S1061-0022-08-00992-8

M3 - Article

AN - SCOPUS:85009738238

VL - 19

SP - 167

EP - 195

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 2

ER -