A note on the action of the Hecke group H(2) on subsets of the form ℚ∗(n)

Abstract

Here, we study the action of the groups [Formula: see text] generated by the linear fractional transformations [Formula: see text] and [Formula: see text] is a positive integer, on the subsets [Formula: see text] and [Formula: see text] is a square-free positive integer. We prove that this action has a finite number of orbits if and only if [Formula: see text] or [Formula: see text]. Moreover, we give an upper bound for the number of orbits for [Formula: see text].