Subgroups of split orthogonal groups over a ring
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).
scientific article; zbMATH DE number 4136227
scientific article