Revisiting the action of a subgroup of the modular group on imaginary quadratic number fields

Abstract

Consider the modular group $\mbox{PSL}(2,\mathbb{Z})=\langle x, \, y \,|\, x^2=y^3=1\rangle$ generated by the transformations $x: z\mapsto -1/z$ and $y:z\mapsto (z-1)/z$. Let $H$ be the proper subgroup $\langle y,\,v\,|\, y^3=v^3=1\rangle$ of $\mbox{PSL}(2,\mathbb{Z})$, where $v=xyx$. The reference (M. Ashiq and Q. Mushtaq, {\em Actions of a subgroup of the modular group on an imaginary quadratic field}, Quasigropus and Related Systems {\bf 14} (2006), 133--146) proposed results concerning the action of $H$ on the subset $\{\frac{a+\sqrt{-n}}{c}\,|\, a,b=\frac{a^2+n}{c}, c \in \mathbb{Z}, c\neq 0\}$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-n})$ for a positive square-free integer $n$. In the current article, the author points out and corrects errors appearing in the aforementioned reference. Most importantly, the corrected estimate for the number of orbits arising from this action is given.