User:Mathemajor/2D Lattices Sublattices of a Given Index

Let $$\Gamma$$ be a two dimensional lattice and $$\Gamma^{\prime}$$ be a sublattice of $$\Gamma$$ such that the index of $$\Gamma^{\prime}$$ in $$\Gamma$$, $$\textstyle \left(\Gamma^{\prime} : \Gamma\right)$$, is $$n$$ and define $$S_{n}$$ as follows:
 * $$S_{n} = \left\{ g = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \in \mbox{ML}_{2}(\mathbb{Z}) \mid ad = n, a > 0, 0 \leq b < d \right\}$$.