On Normal Subgroups of the Group Representation of the Cayley Tree

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.