Orders of products of horizontal class transpositions

TL;DR

This paper investigates the orders of products of horizontal class transpositions within the group $CT(bZ)$, providing a classification of possible product orders and addressing a question from the Kourovka notebook.

Contribution

It characterizes the possible orders of products of horizontal class transpositions in the subgroup $CT_{ ext{infty}}$, partially answering a question posed by S. Kohl.

Findings

Orders of products are in {1,2,3,4,6,12}.

Every number in this set occurs as a product order.

Provides a partial answer to a problem in the Kourovka notebook.

Abstract

The class transposition group $CT(\mathbb{Z})$ was introduced by S. Kohl in 2010. It is a countable subgroup of the permutation group $Sym(\mathbb{Z})$ of the set of integers $\mathbb{Z}$. We study products of two class transpositions $CT(\mathbb{Z})$ and give a partial answer to the question 18.48 posed by S. Kohl in the Kourovka notebook. We prove that in the group $CT_{\infty}$, which is a subgroup of $CT(\mathbb{Z})$ and generated by horizontal class transpositions, the order of the product of a pair of horizontal class transpositions belongs to the set $\{1,2,3,4,6,12\}$, and any number from this set is the order of the product of a pair of horizontal class transpositions.