Generators of the Möbius Group
The Möbius group is generated by for , for , and .
Proposition 5.10
is generated by the set of elements of the following 3 forms:
- , where
- , where
Proof
For , let , , and . Compose a transformation of the form to send to , then a translation to send the image of to , and finally a scaling to send the image of to . The resulting transformation sends to , so the three-point lemma shows it is . Thus is a product of the three stated types and their inverses.
Related
Stated in
- Proposition 5.10§5.2 The Möbius Group