On normal subgroups of the group representation of the Cayley tree

F. H. Haydarov<sup>ab

a</sup> V. I. Romanovskiy Institute of Mathematics, 9 University St., Tashkent 100174, Uzbekistan
^b National University of Uzbekistan named after Mirzo Ulugbek, 4 University St., Tashkent 100174, Uzbekistan

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.

Key words: Cayley tree, $G_{k}$-group, subgroups of finite index, abelian group, homomorphism.

UDC: 512.54

MSC: 20B07, 20E06

Received: 22.06.2022

Language: English

DOI: 10.46698/l0184-0874-2706-y