Sharp three-transitivity of the Möbius group

Statement

For any two ordered triples

(z_1,z_2,z_3),\qquad (w_1,w_2,w_3)

of distinct points of the Riemann sphere, there is a unique Möbius transformation $T$ satisfying

T(z_i)=w_i\qquad(i=1,2,3).

Thus the action of the Möbius group on the sphere is sharply three-transitive.

Proof

There is a Möbius transformation carrying each ordered triple to $(1,0,\infty)$ , given by its cross-ratio coordinate. Composing one normalization with the inverse of the other gives existence. A Möbius transformation fixing three distinct points is the identity, which gives uniqueness.

Quadruple invariant

Once the images of three points are fixed, the image of a fourth is determined by cross-ratio invariance. Consequently, two ordered quadruples of distinct points are in the same Möbius orbit exactly when their cross-ratios agree.

References

Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983. Publisher record. Relevant: Chapter 3, §§1–2.