19.45 (2018)

Let $G$ be a group generated by 3 class transpositions (see the definition in 17.57), and let $m$ be the least common multiple of the moduli of the residue classes interchanged by the generators of $G$. Assume that $G$ does not setwisely stabilize any union of residue classes modulo $m$ except for $\varnothing$ and $\mathbb{Z}$, and assume that the integers $0, 1, \dots, 42$ all lie in the same orbit under the action of $G$ on $\mathbb{Z}$. Is the action of $G$ on $\mathbb{N} \cup \{0\}$ necessarily transitive?

The bound 42 cannot be replaced by a smaller number, since the finite group $\langle \tau_{0(2), 1(2)}, \tau_{0(3), 2(3)}, \tau_{0(7), 6(7)} \rangle$ acts transitively on the set $\{0, \dots, 41\}$, as well as on the set of residue classes modulo 42.