Find an operation so that it makes $D^{2}=\{z\in \mathbb{C}\mid |z|<1\}$ a group.

Question

Let $D^{2}=\{z\in \mathbb{C}\mid |z|<1\}$. Find operation $\oplus: D^{2}\times D^{2}: \rightarrow D^{2}$ so that it makes $D^{2}$ a group.

I have already defined some operations but these don't verify associative property. For example, I defined $f(z)=\frac{z}{1+|z|}$ and I defined $z\oplus w:=f(f(z)+f(w))$, but they don't verify associative property.

Thanks, any help is appreciated.

Answer 1

Choose any bijection $f:D^2\to\mathbb R^2$ (it may even be a diffeomorphism) and let $$z\oplus w:=f^{-1}(f(z)+f(w)).$$