Presentations of SL₂(Z) and PSL₂(Z)

Let $\textrm{SL}_2({\bf Z})$ be the group of $2 \times 2$ matrices with entries in ${\bf Z}$ and with determinant 1. The center of $\textrm{SL}_2({\bf Z})$ is the group of order 2 generated by the scalar matrix $-I_2$, where I ₂ is the unit matrix. The quotient group

$$\textrm{PSL}_2({\bf Z}) = \textrm{SL}_2({\bf Z})/\langle -I_2 \rangle$$

is called the modular group ; it can be identified with the group of rational functions on ${\bf C}$ of the form $(az+b)/(cz+d)$, where a, b, c, d are integers such that $ad - bc = 1$.