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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
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