On The Power Subgroups of The Modular Group

Abstract

We show that the power subgroups $M^{6k}$ ($k>1$) of the Modular group $M=\text{PSL}(2,\mathbb{Z})$ are subgroups of the groups $M^{\prime }(6k,6k)$. Here, the groups $M^{\prime }(6k,6k)$ ($k>1$) are subgroups of the commutator subgroup $M^{\prime }$ of $M$ of index $36k^2$ in $M^{\prime }$.