A group of linear-fractional transformations of the upper half-plane Show that the transformations of the upper half-plane $\mathcal H = \{ x+ \Bbb i y | x, y \in \Bbb R, y>0 \}$ of the form $z \mapsto z' = \frac {az+b} {cz+d}, \ a,b,c,d \in \Bbb Z$ with $ad-bc = 1$ form a group under composition, isomorphic to $SL_2(\Bbb Z)/\{\pm Id\}$.
Deduce that this group is generated by the transformations $z \mapsto -\frac 1 z$ and $z \mapsto z+1$.
Here I showed that it forms a group under compostion. How to show isomorphism and the last part? Thanks.
 A: Let $G$ be that group of transformations. Show ${\rm SL}_2(\Bbb Z)\to G$ via $[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}]\mapsto \frac{az+b}{cz+d}$ is a group homomorphism. Show it is onto, compute its kernel, invoke first isomorphism theorem.
For the second part it suffices to show $[\begin{smallmatrix} 0 & -1 \\ 1 & \phantom{-}0 \end{smallmatrix}]$ and $[\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}]$ generate ${\rm SL}_2(\Bbb Z)$. The exercise almost sounds like it's expecting you to know that these matrices generate ${\rm SL}_2(\Bbb Z)$ so then all you have to do is explain why that proves the claim, but in case you should also want to know why they generate ${\rm SL}_2(\Bbb Z)$ I'll give some ideas on that too.
It suffices to show that given any matrix in ${\rm SL}_2(\Bbb Z)$, we can multiply it on the left by these generators repeatedly until we get the identity matrix.
Think of what these two linear transformations do to $\Bbb Z^2\subset\Bbb R^2$ geometrically. One rotates the plane by a right angle counterclockwise, the other shears a vector horizontally by its height. One can use the latter to ensure the vector's horizontal component is $<$ its height, and then rotate to swap their relative sizes, which keeps reducing (say) the "taxicab" norm $|x|+|y|$ of the vector. Show that if the vectors coordinates to begin with were coprime, this process can be used to take it to the coordinate vector $e_1$.
Since multiplying two matrices $AB$ in effect multiplies each column of $B$ on the left by $A$, we may use this process to keep multiplying a matrix in ${\rm SL}_2(\Bbb Z)$ on the left by these generators and end up with $e_1$ in the first column - since we're still in ${\rm SL}_2(\Bbb Z)$ what must the second column be?
