Groups of Class Transpositions with Prescribed Prime Divisors of the Moduli

Abstract

For a set $\mathcal{P}$ of odd primes, let ${\mathrm{CT}}_{\mathcal{P}}(\mathbb{Z})$ denote the group generated by all class transpositions whose moduli have no odd prime divisors outside $\mathcal{P}$. We prove that \[ \bigl\langle \operatorname{CT}_{\mathcal P_1}(\mathbb{Z}),\operatorname{CT}_{\mathcal P_2}(\mathbb{Z}) \bigr\rangle = \operatorname{CT}_{\mathcal P_1\cup\mathcal P_2}(\mathbb{Z}) \] for any sets ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ of odd primes. This gives a negative answer to Question 21.75 in the Kourovka Notebook.