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On the subgroups of split orthogonal groups. (Russian) Zbl 0647.20043

Let K be a field of odd characteristic, \(GL(2\ell +1,K)\), \(SL(2\ell +1,K)\) the general linear group and the special linear group respectively of degree \(n\) over \(K\), \(D(2\ell +1,K)\) the group of diagonal matrices in \(GL(2\ell +1,K)\). For \(a\in GL(2\ell +1,K)\) let \(a_{ij}\) be the element which lies in the \(i\)-th row and \(j\)-th column and \(a^ t\) the transposed matrix to \(a\). It will be useful to give to rows and columns the following numbers \(1,...,\ell,0,-\ell,...,-1\). Let \(f_{\ell}\in GL(2\ell +1,K)\) be the matrix such that \(f_{ij}=\delta_{i,-j}\) where \(\delta_{kt}\) is the Kronecker symbol. Put \(SO(2\ell +1,K)=\{a\in SL(2\ell +1,K)|\) \(aFa^ t=F\}\), where \[ F=\begin{pmatrix} 0 & 0 & fe \\ 0 & 2 & 0 \\ fe & 0 & 0 \end{pmatrix}, \] \(GO(2\ell+1,K) = \{a\in GL(2\ell +1,K)|\) \(aFa^ t=\lambda F\), \(\lambda \in K^*\}\). Recall that an \(n\times n\) table \(\sigma =(\sigma_{ij})\) of ideals of \(K\) is called a D-net of ideals in \(K\) of order \(n\) if \(\sigma_{ir}\sigma_{rj}\subset \sigma_{ij}\) and \(\sigma_{ii}=K\) for any i,j,r. Let \((\sigma)\) be a D-net of ideals in K of order \(2\ell +1\), then \((\sigma)\) is called an orthogonal D-net if \(\sigma_{ij}=\sigma_{-j,-i}\) for any \(i\), \(j\) and \(\sigma_{i,- i}=\sum \sigma_{ij}\sigma_{j,-i}\), \(1\leq j\leq \ell\), \(j=\pm i\) for any \(i\). Let \(\sigma\) be an orthogonal D-net, \(SO(2\ell +1,K)(\sigma)=SL(2\ell +1,K)\cap GL(2\ell +1,K)(\sigma)\), where \(GL(2\ell +1,K)(\sigma)=\{a\in GL(2\ell +1,K)|\) \(a_{ij}=\delta_{ij} (mod \sigma_{ij})\}\) and \(N_{SO(2\ell +1,K)}(\sigma)\), \(N_{GO(2\ell +1,K)}(\sigma)\) are the normalizers of \(GL(2\ell +1,K)(\sigma)\) in \(SO(2\ell +1,K)\) and \(GO(2\ell +1,K)\) respectively.
Then the main result of the paper under review is the following Theorem: Let \(K\) be a field of odd characteristic consisting of at least thirteen elements. Then for any subgroup \(H\) of \(SO(2\ell +1,K)\) containing the subgroup \(D(2\ell +1,K)\cap SO(2\ell +1,K)\) there exists a unique D-net \(\sigma\) of ideals in \(K\) of order \(2\ell +1\) such that \(SO(2\ell +1,K)(\sigma)\leq H\leq N_{SO(2\ell +1,K)}(\sigma)\).
Reviewer: V.Yanchevskij

MSC:

20G15 Linear algebraic groups over arbitrary fields
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
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