I have been given the task of showing that the function $\Lambda:SL(2)\rightarrow Isom(\Bbb H_2)$ defined by $\Lambda(g)(z)=\frac{g_{(1,1)}z+g_{(1,2)}}{g_{(2,1)}z+g_{(2,2)}},g\in SL(2),z\in \Bbb H_2$ is a surjective group homomorphism. However, I'm uncertain on how to classify the group operation of the isometries of the hyperbolic plane? I have tried reading through my notes from university and some articles online, but I can't seem to understand what the group operation actually is.
Furthermore, how do elements of the form $\frac{az+b}{cz+d}, z\in \Bbb H_2, ad-bc=1$ describe elements in $Isom(\Bbb H_2)$?
I apologize if these questions are too broad or misguided... (I guess that shows the state of my confusion).
The element of Isom is the map
$$ z \mapsto \frac{az + b}{cz + d} $$ where I've omitted all the g's and their subscripts. The group operation is "composition of functions." The identity function (which is $\Lambda(I)$, by the way) serves as the identity element for the composition operation.