Transvection modules in the overgroups of a non-split maximal torus

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2014.3.10228 Transvection modules in the overgroups of a non-split maximal torus Koibaev, V. A. , Dzhusoeva, N. A. Vladikavkaz Mathematical Journal 2014. Vol. 16. Issue 3. Abstract: The aim of this article is to investigate the modules of transvections and rings of multipliers subgroups of the general linear group \(G=GL(n,k)\) of degree \(n\) over a field \(k\), containing non-split maximal torus \(T=T(d)\), associated with a radical extension of \(k(\sqrt[n]{d})\) of the degree \(n\) of the ground field \(k\) of an odd characteristic (minisotropic torus). We find a full list of \(2\cdot[(\frac{n-1}{2})^{2}]\) relations (\([\cdot]\) - integer part) of the modules of transvections. We prove that all ring of multipliers coincide, and all modules transvections are ideals of the ring of multipliers. All results were proved by the assumption that the ground field \(k\) is the field of fractions of a principal ideal domain. Keywords: overgroups, intermediate subgroups, non-split maximal torus, transvection, module transvections Language: Russian Download the full text For citation: Koibaev V. A., Dzhusoeva N. A. Transvection modules in the overgroups of a non-split maximal torus // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 16, no. 3, pp. 3-8. DOI 10.23671/VNC.2014.3.10228 + References 1. Borevich Z. I., Kojbaev V. A. O kol'cah mnozhitelej, svjazannyh s promezhutochnymi podgruppami dlja kvadratichnyh torov. Vestn. SPbGU [Vestnik of Saint Petersburg University], 1993, vol. 1, no. 2, pp. 5-10 (Russian). 2. Borevich Z. I., Kojbaev V. A., Chan Ngok Hoj. Lattices of subgroups in GL(2,Q) that contain a nonsplittable torus. J. Soviet Math., 1993, vol. 63, no. 6, pp. 622-633. 3. Dzhusoeva N. A. Net rings that are normalizable by a nonsplit maximal torus. Trudy Inst. Mat. i Mekh. UrO RAN [Proc. of the Steklov Institute of Math.], vol. 19, no. 3, pp. 113-119 (Russian). 4. Dzhusoeva N. A., Kojbaev V. A. Maximal Subgroups that Contain a Torus and are Related to the Field of Fractions of a Dedekind Domain. J. Math. Sci., 2004, vol. 124, no. 1, pp. 4763-4765. 5. Kojbaev V. A. Subgroups of the group GL(2,Q) that contain a nonsplittable maximal torus. Soviet Math. Dokl., 1990, vol. 41, no. 3, pp. 414-416. 6. Kojbaev V. A. Transvection in the subgroups of the general linear group containing a nonsplit maximal torus. St. Petersburg Math. J., 2010, vol. 21, no. 5, pp. 731-742. 7. Leng S. Algebra [Algebra], Moskva, Nauka, 1968, 572 p. (Russian). The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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