Geodesic
On normal subgroups of the group representation of the Cayley tree
Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 4, pp. 135-142 
F. H. Haydarov. On normal subgroups of the group representation of the Cayley tree. Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 4, pp. 135-142. http://geodesic.mathdoc.fr/item/VMJ_2023_25_4_a11/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Haydarov, F. H., “New Normal Subgroups for the Group Representation of the Cayley Tree”, Lobachevskii Journal of Mathematics, 39:2 (2018), 213–217 | DOI | MR | Zbl

[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, 5:2 (2014), 234–240 | MR | Zbl

[3] Cohen D. E. and Lyndon, R. C., “Free Bases for Normal Subgroups of Free Groups”, Transactions of the American Mathematical Society, 108 (1963), 526–537 | DOI | MR | Zbl

[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, 79 (2006), 399–407 | DOI | MR | Zbl

[5] Rozikov, U. A., Gibbs Measures on a Cayley Tree, World Scientific, Singapore, 2013 | DOI | MR

[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, 7:2 (2013), 323–355 | DOI | MR

[7] Rozikov, U. and Haydarov, F., “Invariance Property on Group Representations of the Cayley Tree and Its Applications”, Results in Mathematics, 77:6 (2022), 241 | DOI | MR | Zbl

[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, 191 (2017), 910–923 | DOI | MR | Zbl

[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, 18:1 (2015), 1–22 | DOI | MR | Zbl

[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, 210 (2022), 261–274 | DOI | MR | Zbl

[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 | MR | Zbl