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

N. A. Vavilov ; V. A. Petrov

Algebra i analiz, Tome 19 (2007) no. 2, pp. 10-51.

Résumé

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 $\operatorname{Ep}(2l,R)$, analogous results appeared in previous papers by the authors (Zapiski Nauchn. Semin. POMI, 2000, v. 272; Algebra i Analiz, 2003, v. 15, no. 3).

Détail

Keywords: General linear group, overgroup, split elementary orthogonal group.

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