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


Authors: N. A. Vavilov and 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
DOI: https://doi.org/10.1090/S1061-0022-08-00992-8
Published electronically: February 1, 2008
MathSciNet review: 2333895
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Abstract: Let $ R$ be a commutative ring with 1, $ n$ a natural number, and let $ l=[n/2]$. 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 $ H$, 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 $ 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 $ \operatorname{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).


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Additional Information

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

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

DOI: https://doi.org/10.1090/S1061-0022-08-00992-8
Keywords: General linear group, overgroup, split elementary orthogonal group
Received by editor(s): November 20, 2003
Published electronically: 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
Article copyright: © Copyright 2008 American Mathematical Society

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