Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 3 / 2D / Solution

Existence follows from the three-transitivity of Möbius transformations: one can send three distinct source points to $0,1,\infty$ and then send those to the three distinct target points.

For uniqueness, suppose $S$ and $T$ have the same values at three distinct points. Then $M=T^{-1}S$ fixes those points. Ifthe finite fixed points satisfy $c{z}^2+(d-a)z-b=0$. A nonidentity Möbius map therefore has at most two fixed points on the Riemann sphere. Since $M$ has three, it is the identity, and $S=T$.