Geodesic
The Number of Finite Index Subgroups of Baumslag--Solitar Groups
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 10 (2010) no. 2, pp. 54-60

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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.
Keywords: Baumslag–Solitar group, the number of finite index subgroups, transitive two generator subgroups of $S_n$. 
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     author = {F. A. Dudkin and V. A. Churkin},
     title = {The {Number} of {Finite} {Index} {Subgroups} of {Baumslag--Solitar} {Groups}},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
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F. A. Dudkin; V. A. Churkin. The Number of Finite Index Subgroups of Baumslag--Solitar Groups. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 10 (2010) no. 2, pp. 54-60. http://geodesic.mathdoc.fr/item/VNGU_2010_10_2_a4/