Possible Duplicate:
How do I find out the symmetry of a function?
Let $f:\mathbf{P}^1 \longrightarrow \mathbf{P}^1$ be a Möbius transformation $z\mapsto (az+b)/(cz+d)$ sending $\{0,1,\infty\}$ to $\{0,1,\infty\}$ with $ad-bc = 1$.
I suspect there are only a finite number of such Möbius transformations. What are these?
A non-trivial example is $f(z) = 1/(1-z)$. It sends $0$ to $1$, $1$ to $\infty$ and $\infty$ to $0$. Note that $f(z) = -z+1$ is not an example, because $ad-bc = -1$ in this case.