Subgroups of split orthogonal groups over a ring
type: Item
instance of: scholarly article
title: Subgroups of split orthogonal groups over a ring
zbMATH Open document ID: 0694.20032
DOI: 10.1007/BF00969861
published in: Siberian Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: In this paper some results on the description of subgroups of the general linear group over an associative ring obtained by the author and others are carried over to the splittable orthogonal group \(\Gamma =GO_ n(R)\) over a commutative ring R with \(2\in R^*\). We will number rows and columns of matrices in the following way: 1,...,\(\ell,-\ell,...,-1\) if \(n=2\ell\) and 1,...,\(\ell,0,-\ell,...,-1\) if \(n=2\ell +1\). Let on the set \(\{\) 1,...,-1\(\}\) some equivalence relation \(\nu\) be fixed. Denote by \(E_{\Gamma}(\nu)\) the subgroup of \(\Gamma\) generated by all elementary orthogonal transvections: \(T_{ij}(\alpha)=e+\alpha e_{ij}-\alpha e_{-i,-j}\) for \(i,j=0\), \(i=\pm j\) and \(T_{i0}(\alpha)=e+\alpha e_{i0}-2\alpha e_{0,-i}-\alpha^ 2e_{i,-i}\) (for \(n=2\ell\), \(T_{i0}(\alpha)\) is absent). Suppose that the equivalence \(\nu\) satisfies the following condition: if \(i\sim j\) then -i\(\sim -j\) and every equivalence class contains \(\geq 5\) elements. It is proved that if R is a Dedekind ring with \(2\in R^*\) and the equivalence relation \(\nu\) satisfies the condition above then for each subgroup H of \(\Gamma\) containing the subgroup \(E_{\Gamma}(\nu)\) the following inclusions hold: \(E_{\Gamma}(\sigma)\leq H\leq N_{\Gamma}(\sigma)\). Here \(\sigma =\{\sigma_{ij}\}\) is some family of ideals of R satisfying the conditions: \(\sigma_{ij}\sigma_{jk}\subseteq \sigma_{ik}\), \(\sigma_{ii}=R\) and \(\sigma_{i,- i}=\sum_{j}\sigma_{ij}\sigma_{j,-i}\) (orthogonal D-network), \(E_{\Gamma}(\sigma)\) is the subgroup of \(\Gamma\) generated by \(T_{ij}(\alpha)\), \(\alpha \in \sigma_{ij}\), \(i,j\in \{1,...,-1\}\), \(N_{\Gamma}(\sigma)\) is the normalizer of the principal congruence subgroup (Theorem 1). If R is a commutative ring, \(2\in R^*\), \(\sigma\) is an orthogonal network of ideals then under some conditions on \(\sigma\) the normalizer \(N_{\Gamma}(\sigma)\) coincides with the normalizer in \(\Gamma\) of \(E_{\Gamma}(\sigma)\) (Theorem 2).
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 4136227
zbMATH Keywords: subgroups
zbMATH Keywords: splittable orthogonal group
zbMATH Keywords: elementary orthogonal transvections
zbMATH Keywords: principal congruence subgroup
zbMATH Keywords: network of ideals
author: Nikolai Vavilov
reviewed by: V. Ya. Bloshchitsyn
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00969861
OpenAlex ID: W2093275040
description: scientific article; zbMATH DE number 4136227