Triple Transitivity of the Möbius Group
For any two triples of distinct points in , there is a unique Möbius transformation sending the first triple to the second.
Lemma 5.8 (Triple Transitivity)
For any triples of distinct points and , there exists a such that for .
Proof
For a triple , construct a Möbius transformation sending it to by , with the cases involving modified appropriately. Construct similarly for . Then sends to . Uniqueness follows from the three-point lemma.
Related
Stated in
- Lemma 5.8 (Triple Transitivity)§5.2 The Möbius Group