On Cyclic Subgroups of a Full Linear Group of Third Degree over a Field of Zero Characteristic

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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.2018.2.14722 On Cyclic Subgroups of a Full Linear Group of Third Degree over a Field of Zero Characteristic Pachev U. M. , Isakova M. M. Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 2. Abstract: In this paper, using the concept of the spectrum of a matrix, we give an explicit form for the elements of any cyclic subgroup in the full linear group \(GL_3(F)\) of the third degree over the field \(F\) of characteristic zero. In contrast to iterative methods, each element of the cyclic subgroup \(\langle M \rangle\) of the group \(GL_3(F)\) is a linear combination of \(M^{0}\), \(M\), \(M^{2}\), with coefficients easily computed using determinants of the third order, composed by certain powers of the eigenvalues of the matrix \(M\). In fact, we offer a new approach based on a property of the characteristic roots of the polynomial of the matrix. Note also that we present a method that involves the previously known eigenvalues of the matrix. Finally, basing on the results about the explicit form of the elements of any cyclic subgroup of the group \(GL_3(F)\) we derive à formula for the cyclic subgroups of prime order \(p\) of linear group \(GL_3(K^{(p)})\) over a circular field \(K^{(p)}\) of characteristic zero that is of interest in their own right in the theory of infinite groups. Keywords: complete linear group, cyclic subgroups, spectrum of a matrix, diagonalizable matrix, \(n\)-circular field, algebraic closure of a field Language: Russian Download the full text For citation: Pachev U. M., Isakova M. M. On cyclic subgroups of a full linear group of third degree over a field of zero characteristic. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 2, pp. 62-68. DOI 10.23671/VNC.2018.2.14722 + References 1. Pachev U. M., Shokuev V. N. Cyclic Subgroups of the Group \(GL_2(F)\). Izv. KBNTs RAN [Izvestia KBSC of RAS], 2001, vol. 7, no. 2, pp. 72-74 (in Russian). 2. Shokuev V. N. Cyclic Subgroups of the Group \(GL_3(F)\). Izv. KBNTs RAN [Izvestia KBSC of RAS], 2001, vol. 7, no. 2, pp. 75-77 (in Russian). 3. Zhemuhova M. Z., Pachev U. M. Cyclic Subgroups of Second Degree Full Linear Group Over a Field of the Zero Characteristic. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], 2011, vol. 13, no. 3, pp. 17-21 (in Russian). 4. Voevodin V. V., Kuznecov Ju. A. Matricy i vychislenija [Matrices and Calculations], Moscow, Nauka, 1984 (in Russian). 5. Lancaster P. Teorija matric [Theory of Matrices], Moscow, Nauka, 1973 (in Russian). 6. Pachev U. M. Cyclic Subgroups of the Group \(GL_3 (F )\) over a Field of Characteristic Zero. Proceedings of the International Scientific Conference "Actual Problems of Applied Mathematics and Physics", Nalchik, Izd-vo IPMA KBNC RAN, 2017, pp. 167-168 (in Russian). 7. Faddeev D. K., Faddeeva V. N. Vychislitel'nye metody linejnoj algebry [Computational Methods of Linear Algebra], Moscow, Fizmatgiz, 1963 (in Russian). 8. Lidl R., Niederreiter H. Konechnye polja [Finite Fields], vol. 1, Moscow, Mir, 1988 (in Russian). 9. Kostrikin I. Vvedenie v algebru [Introduction to Algebra], Moscow, Nauka, 1977 (in 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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