The modular group is the group $G$ consisting of all linear fractional transformations $\phi$ of the form $$\phi(z)=\frac{az+b}{cz+d}$$ where $a,b,c,d$ are integers and $ad-bc=1$. I have read that $G$ is generated by the transformations $\tau(z)=z+1$ and $\sigma(z)=-1/z$. Is there an easy way to prove this? In particular, is there a proof that uses the relation between linear fractional transformations and matrices? Any good reference would be helpful.
Thank you, Malik