21.8 (2026)

As in 17.57, let $r(m) = \{r + km \mid k \in \mathbb{Z}\}$ for integers $0 \leqslant r < m$; for $r_1(m_1) \cap r_2(m_2) = \varnothing$ let the class transposition $\tau_{r_1(m_1), r_2(m_2)}$ be the involution which interchanges $r_1 + tm_1$ and $r_2 + tm_2$ for each integer $t$ and fixes everything else, and let $\text{CT}(\mathbb{Z})$ be the group generated by all class transpositions.

Let $\text{CT}_k$ be the subgroup of $\text{CT}(\mathbb{Z})$ generated by the class transpositions $\tau_{r_1(k), r_2(k)}$ for $0 \leqslant r_1 \neq r_2 < k$. Since $\tau_{r_1(k), r_2(k)}$ permutes the residue classes modulo $k$, the group $\text{CT}_k$ is isomorphic to the symmetric group $S_k$. Let $\text{CT}_{(k)} = \langle \text{CT}_2, \text{CT}_3, \dots, \text{CT}_k \rangle$. Is it true that for $k > 3$ the group $\text{CT}_{(k)}$ is isomorphic to the symmetric group $S_N$, where $N$ is the least common multiple of the numbers $2, 3, \dots, k$?