What is the meaning of 'Analytic Isomorphism'?

Question

I am reading the book "Problems in the Theory of Modular Forms: M. Ram Murty" Chapter-3, Exercise 3.1.4. In this, I need to show that the map $$ z \mapsto \frac{z-i}{z+i} $$ is an analytic isomorphism from the upper half-plane $\mathbb{H}$ to the unit disk $$\mathbb{D}=\{w \in \mathbb{C}: |w|<1\}.$$ Since the map is defined on the upper half-plane which is not a group with respect to addition or multiplication, in what sense, we call the above map to be an 'Isomorphism'? Is it just a set bijection? Any clarification in this regard would be appreciated!

Answer 1

It means that the map is bi-holomorphic. The map is the well-known "Cayley-transform". There is a generalization $\Phi\colon G\rightarrow {\rm Lie}(G)$ defined on groups, if you want groups, namely on certain algebraic groups and Lie groups. For this see the interesting article by Kostant and Michor.