A note on the horizontal class transposition group

TL;DR

This paper proves that for integers greater than 3, the horizontal class transposition group is isomorphic to a symmetric group, solving a conjecture from the Kourovka notebook.

Contribution

It establishes the isomorphism between the horizontal class transposition group and a symmetric group for all n > 3, confirming a conjecture in group theory.

Findings

Proves $CT_{(n)} \\cong S_N$ for $n > 3$

Solves a conjecture from the Kourovka notebook

Provides a group-theoretic characterization of $CT_{(n)}$

Abstract

Let $n$ be an integer with $n > 1$. For every $r$ satisfying the inequalities $0 \leq r < n$, the residue class modulo $n$ is defined as $r(n)=\{r + kn | k \in Z\}$, where $Z$ is the set of all integers. Then for $0 \leq r_1\neq r_2 < n$, the horizontal class transposition $\tau_{r_1(n), r_2(n)}$ is an involution that interchanges $r_1 + kn$ and $r_2 + kn$ for each integer $k$ and fixes everything else. The horizontal class transposition group $CT_n$ is generated by all horizontal class transposition $\tau_{r_1(n), r_2(n)}$. Let $N$ be the least common multiple of the numbers $2, 3, . . . , n$ and $CT_{(n)}=\langle CT_2,CT_3,...,CT_n\rangle$. In this note, we prove that for $n>3$, $CT_{(n)}\cong S_N$, where $S_N$ is the symmetric group of degree $N$. Thus, we solve a conjecture proposed by Bardakov and Iskra, which has been included in the kourovka notebook: Unsolved problems in group theory, Novosibirsk, 2026.