Cyclical Elementary Nets

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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.2017.1.5818 Cyclical Elementary Nets Dzhusoeva, N. A. , Dryaeva R. Y. Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 1. Abstract: Let \(R\) be a commutative ring with the unit and \(n\in\Bbb{N}\). A set \(\sigma = (\sigma_{ij})\), \(1\leq{i, j} \leq{n},\) of additive subgroups of the ring \(R\) is a net over \(R\) of order \(n\), if \( \sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}} \) for all \(1\leq i, r, j\leq n\). A net which doesn't contain the diagonal is called an elementary net. An elementary net \(\sigma = (\sigma_{ij}), 1\leq{i\neq{j} \leq{n}}\), is complemented, if for some additive subgroups \(\sigma_{ii}\) of \(R\) the set \(\sigma = (\sigma_{ij}), 1\leq{i, j} \leq{n}\) is a full net. An elementary net \(\sigma\) is called closed, if the elementary group \( E(\sigma) = \langle t_{ij}(\alpha) : \alpha\in \sigma_{ij}, 1\leq{i\neq{j}} \leq{n}\rangle \) doesn't contain elementary transvections. It is proved that the cyclic elementary odd-order nets are complemented. In particular, all such nets are closed. It is also shown that for every odd \(n\in\Bbb{N}\) there exists an elementary cyclic net which is not complemented. Keywords: intermediate subgroup, non-split maximal torus, net, cyclic net, net group, elementary group, transvection. Language: Russian Download the full text For citation: Dzhusoeva N. A., Dryaeva R. Y. Cyclical elementary nets. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp.26-29. DOI 10.23671/VNC.2017.1.5818 + References 1. Koibaev V. A., Shilov A. V. On subgroups of a full linear group, which contains non-split maximal torus. Zap. Nauch. Seminarov POMI RAN. 2010, vol. 375, pp. 130-139. 2. Borevich Z. I. On subgroups of a full linear group, which is rich by transvections. Zap. Nauch. Seminarov LOMI. 1978, vol. 75, p. 22-31. 3. Kargapolov M. I., Merzlyakov Y. I. The Fundamental of the Group Theory. Nauka, M., 1982. 288 p. 4. Levchuk V. M. On some remarks about L. Dikson theorem. Algebra and Logic, 1983, vol. 22, no. 5, pp. 504-517. 5. The "Kourovsk notebook". Open Problems of the Group Theory. Novosibirsk, 2010. Vol. 17. 6. Koibaev V. A. Elementary nets and linear groups. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2011, vol. 17, no. 4, p. 134-141. 7. Koibaev V. A. Transvections and subgroups of the full linear group, which contains non-split torus. St. Petersburg Math. J., 2009, vol. 21, no. 5, p. 70-86. 8. Koibaev V. A. Nets, associated with elementary nets. Vladikavkazskij Matematicheskij Zhurnal [Vladikavkaz Math. J.], 2010, vol. 12, no. 4, p. 39-43. 9. Koibaev V. A., Nuzhin Y. N. Subgroups of the Chevalley Groups and Lie Rings Definable by a Collection of Additive Subgroups of the Initial Ring. J. of Math. Sci., 2014, vol. 18, no. 1, pp. 75-84. 10. Dryaeva R. Y., Koibaev V. A. Decomposition of elementary transvections and elementary groups. Zap. Nauch. Semin. POMI RAN, 2015, vol. 435, pp. 33-40. 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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