Group Properties of Extended Mobius Transformations

Ifis a Mobius transformation, then the domain issince ifthe denominator is zero. Ifthen the domain is

We can extend Mobius transformations to the whole ofincluding the point at infinity by introducing the extended complex plane

We define the extended Mobius transformationby

We can define the extended Mobius transformationby

We can test for the group properties – identity, closure, inverse and associativity – one by one.

1. The identity is an extended Mobius transformation with a=1 and b=c=d=0 . Note that the identity sends the point at infinity to the point at infinity .

2. Ifandthenwhich is a Mobius transformation so the closure axiom is satisfied.

3. Ifthenso the inverse axiom is satisfied.

4. Associativity follows from the general property of composition of functions.

All the group axioms are satisfied so the set of extended Mobius transformations is a group.