This article is cited in 1 paper

The Number of Finite Index Subgroups of Baumslag–Solitar Groups

F. A. Dudkin^a, V. A. Churkin<sup>ba

a</sup> Novosibirsk State University
^b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: 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 $S_n$.

UDC: 512.543

Received: 17.03.2010

 English version: Journal of Mathematical Sciences, 2012, 186:3, 387–393