1. The composition of two Möbius transformations is a Möbius transformation, and so is the inverse.
2. Given three distinct points \(z_1, z_2, z_3\) and three distinct points \(w_1, w_2, w_3\), there exists a unique Möbius transformation \(f : \hat{\mathbb{C}} \mapsto \hat{\mathbb{C}}\) that maps \(z_j\) to \(w_j , j = 1, 2, 3\):

\[ \begin{aligned} f_2^{-1} \circ f_1 \end{aligned} \]

where \(f_1\) maps \(z_1, z_2, z_3\) to \(0, 1, \infty\), and \(f_2\) maps \(w_1, w_2, w_3\) to \(0, 1, \infty\).

For example, what is the Möbius transformation that maps \(0\) to \(-1, i\) to \(0\), and \(\infty\) to 1 ?

More at Analysis of a Complex Kind.