The Number of Finite Index Subgroups of Baumslag–Solitar Groups

Gelman established a simple formula for the number of finite index subgroups of Baumslag–Solitar groups $BS(p,q)=\langle a,\ t\ | \ t^{-1}a^pt=a^q \rangle$, where $p$ and $q$ are co-prime integers. In this paper we give a generalization of this formula for arbitrary nonzero integers. The proof was obtained by calculating the number of permutations $y\in S_n$ such that subgroup of $S_n$ generated by $x$ and $y$ is transitive, where $x\in S_n$ is given.