Overgroups of 𝐸𝑂(𝑛,𝑅)

Vavilov, N.; Petrov, V.

doi:10.1090/S1061-0022-08-00992-8

Overgroups of $\mathrm {EO}(n,R)$
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by N. A. Vavilov and V. A. Petrov
Translated by: N. A. Vavilov

St. Petersburg Math. J. 19 (2008), 167-195

DOI: https://doi.org/10.1090/S1061-0022-08-00992-8

Published electronically: February 1, 2008

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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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