Orders of products of slanted class transpositions

Abstract

We study the orders of products of two class transpositions in the group CT ⁡ ( Z ) , a simple subgroup of the symmetric group on the integers. For pairs of class transpositions sharing a common vertex, we prove that the order of their product is either 1, 3, or ∞, and provide a precise criterion for the infinite order case. Furthermore, we investigate pairs of equal-residue and equal-modulus class transpositions, establishing conditions under which their product has finite or infinite order. Our results provide a partial answer to a question posed in the Kourovka notebook (see Question 18.48).