St. Petersburg Mathematical Society

Author(s): N. A. Vavilov; V. A. Petrov
Translated by: N. A. Vavilov
Original publication: Algebra i Analiz, tom 19 (2007), nomer 2.
Journal: St. Petersburg Math. J. 19 (2008), 167-195.
MSC (2000): Primary 20G35
Posted: February 1, 2008
Retrieve article in: PDF

Abstract: Let

be a commutative ring with 1,

a natural number, and let

. Suppose that $2\in R^*$ and $l\ge 3$ . We describe the subgroups of the general linear group $\operatorname{GL}(n,R)$ that contain the elementary orthogonal group $\operatorname{EO}(n,R)$ . The main result of the paper says that, for every intermediate subgroup

, there exists a largest ideal $A\trianglelefteq R$ such that $\operatorname{EEO}(n,R,A)= \operatorname{EO}(n,R)E(n,R,A)\trianglelefteq H$ . Another important result is an explicit calculation of the normalizer of the group $\operatorname{EEO}(n,R,A)$ . If

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 $\operatorname{EO}(2l,R)$ and the elementary symplectic group

, analogous results appeared in previous papers by the authors (Zapiski Nauchn. Semin. POMI, 2000, v. 272; Algebra i Analiz, 2003, v. 15, no. 3).

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N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii Pr. 28, Staryi Peterhof, St. Petersburg 198504, Russia

V. A. Petrov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii Pr. 28, Staryi Peterhof, St. Petersburg 198504, Russia

DOI: 10.1090/S1061-0022-08-00992-8
PII: S 1061-0022(08)00992-8
Keywords: General linear group, overgroup, split elementary orthogonal group
Received by editor(s): 20/NOV/2003
Posted: February 1, 2008
Additional Notes: The present paper has been written in the framework of the RFBR projects 01-01-00924 St.-Petersburg State Univ.), 03-01-00349 (POMI RAN), INTAS 00-566 and INTAS 03-51-3251. At the final stage the work of the authors was supported by the express grant of the Russian Ministry of Higher Education ``Overgroups of semisimple groups'' E02-1.0-61
Copyright of article: Copyright 2008, American Mathematical Society

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