Orders of Products of Slanted Class Transpositions

TL;DR

This work extends the study of orders of products of class transpositions from horizontal to slanted transpositions in the group $CT(\mathbb{Z})$, using a graph- theoretic framework $\Gamma(\tau_1,\tau_2)$ to connect combinatorial structure with permutation orders. For pairs sharing a common vertex, the authors show $ord(\tau_1\tau_2)\in\{1,3,\infty\}$, with $\infty$ exactly when the residue classes intersect but are not identical. In the equal-residue and equal-modulus regimes, they establish infinite order in many configurations (and orders in $\{1,2,3,6,\infty\}$ under a ratio constraint), supported by Diophantine constructions and component-length arguments. Together, these results contribute to the finiteness claim for the set of possible orders and provide explicit criteria to determine when a given slanted-pair product has finite or infinite order, enhancing understanding of generalized class transpositions.

Abstract

In the present work, we continue the research initiated in the preprint: V. G. Bardakov, A. L. Iskra, Orders of products of horizontal class transpositions, arXiv:2409.13341, and related to S. Kohl's question on the orders of products of pairs of class transpositions. In the preprint, an answer was given to S. Kohl's question for horizontal class transpositions. In the present work, the products of pairs of slanted class transpositions are considered, and under certain conditions, their orders are determined, with it being established that the number of different orders is finite.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Theorem 2.2
• Theorem 3.1
• proof
• Corollary 3.2
• Proposition 3.3
• proof
• Definition 4.1