Orders of products of slanted class transpositions

Abstract We study the orders of products of two class transpositions in the group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>CT</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>Z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{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).