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DOI: 10.23671/VNC.2015.2.7272 The net and elementary net group associated with non-split maximal torus
Dzhusoeva, N. A.
Vladikavkaz Mathematical Journal 2015. Vol. 17. Issue 2.
Abstract:
The elements of matrixes of a non-split maximal torus \(T=T(d)\) (associated with a radical extension \(k(\sqrt[n]{d})\) of degree \(n\) of the ground field \(k\)) generate some subring \(R(d)\) of the field \(k\). Let \(R\) be an intermediate subring, \(R(d)\subseteq{R}\subseteq{k}\), \(d\in{R}\), \(A_1\subseteq\dots\subseteq A_n\) be a chain of ideals of the ring \(R\), and \( d A_n\subseteq A_1.\) By \(\sigma = (\sigma_{ij})\) we denote the net of ideals defined by \(\sigma_{ij}= A_{i+1-j}\) with \( j\leq i\) and \(\sigma_{ij}=dA_{n+i+1-j}\) with \(j\geq i+1\). By \(G(\sigma)\) and \(E(\sigma)\) we denote the net and the elementary net group, respectively. It is proved, that \(TG(\sigma)\) and \(TE(\sigma)\) are intermediate subgroups of \(GL(n, k)\) containing the torus \(T\).
Keywords: overgroup, intermediate subgroup, elementary group, non-split maximal torus, transvection
Language: Russian
For citation: Dzhusoeva N. A. The net and elementary net group associated with non-split maximal torus. Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 17, no. 2, pp.12-15.
DOI 10.23671/VNC.2015.2.7272
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