On Normal Subgroups of the Group Representation of the Cayley Tree

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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: 10.46698/l0184-0874-2706-y On Normal Subgroups of the Group Representation of the Cayley Tree Haydarov, F. H. Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 4. Abstract: Gibbs measure plays an important role in statistical mechanics. On a Cayley tree, for describing periodic Gibbs measures for models in statistical mechanics we need subgroups of the group representation of the Cayley tree. A normal subgroup of the group representation of the Cayley tree keeps the invariance property which is a significant tool in finding Gibbs measures. By this occasion, a full description of normal subgroups of the group representation of the Cayley tree is a significant problem in Gibbs measure theory. For instance, in [1, 2] a full description of normal subgroups of indices four, six, eight, and ten for the group representation of a Cayley tree is given. The present paper is a generalization of these papers, i. e., in this paper, for any odd prime number \(p\), we give a characterization of the normal subgroups of indices \(2n\), \(n\in\{p, 2p\}\) and \(2^i, i\in \mathbb{N},\) of the group representation of the Cayley tree. Keywords: Cayley tree, \(G_{k}\)-group, subgroups of finite index, abelian group, homomorphism Language: English Download the full text For citation: Haydarov, F. H. On Normal Subgroups of the Group Representation of the Cayley Tree, Vladikavkaz Math. J., 2023, vol. 25, no. 4, pp. 135-142. DOI 10.46698/l0184-0874-2706-y + References 1. Haydarov, F. H. New Normal Subgroups for the Group Representation of the Cayley Tree, Lobachevskii Journal of Mathematics, 2018, vol. 39, no. 2, pp. 213-217. DOI: 10.1134/S1995080218020142. 2. Rozikov, U. A. and Haydarov, F. H. Normal Subgroups of Finite Index for the Group Represantation of the Cayley Tree, TWMS Journal of Pure and Applied Mathematics, 2014, vol. 5, no. 2, pp. 234-240. 3. Cohen D. E. and Lyndon, R. C. Free Bases for Normal Subgroups of Free Groups, Transactions of the American Mathematical Society, 1963, vol. 108, pp. 526-537. 4. Normatov, E. P. and Rozikov, U. A. A Description of Harmonic Functions via Properties of the Group Representation of the Cayley Tree, Mathematical Notes, 2006, vol. 79, pp. 399-407. DOI: 10.1007/s11006-006-0044-4. 5. Rozikov, U. A. Gibbs Measures on a Cayley Tree, Singapore, World Scientific, 2013. DOI: 10.1142/8841. 6. Levgen, V. B., Natalia, V. B., Said, N. S. and Flavia, R. Z. On the Conjugacy Problem for Finite-State Automorphisms of Regular Rooted Trees, Groups, Geometry, and Dynamics, 2013, vol. 7, no. 2, pp. 323-355. DOI: 10.4171/GGD/184. 7. Rozikov, U. and Haydarov, F. Invariance Property on Group Representations of the Cayley Tree and Its Applications, Results in Mathematics, 2022, vol. 77(6), Article no. 241. DOI: 10.1007/s00025-022-01771-9. 8. Rozikov, U. A. and Haydarov F. H. Four Competing Interactions for Models with an Uncountable Set of Spin Values on a Cayley Tree, Theoretical and Mathematical Physics, 2017, vol. 191, pp. 910-923. DOI: 10.1134/S0040577917060095. 9. Rozikov, U. A. and Haydarov, F. H. Periodic Gibbs Measures for Models with Uncountable Set of Spin Values on a Cayley Tree, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2015, vol. 18, no. 1, pp. 1-22. DOI: 10.1142/S021902571550006X. 10. Haydarov, F. H. and Ilyasova, R. A. On Periodic Gibbs Measures of the Ising Model Corresponding to New Subgroups of the Group Representation of a Cayley Tree, Theoretical and Mathematical Physics, 2022, vol. 210, pp. 261-274. DOI: 10.1134/S0040577922020076. 11. Malik, D. S, Mordeson, J. N. and Sen, M. K. Fundamentals of Abstract Algebra, McGraw-Hill Com., 1997. 12. Rose, H. E. A Course on Finite Groups, Springer Science and Business Media, 2009. 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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