Conformal automorphism of Riemann sphere equals fractional linear transformation

This article describes the computation of the conformal automorphism group of a domain or a Riemann surface
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Statement

Any conformal automorphism of the Riemann sphere is a fractional linear transformation, i.e. a map of the form:

$z\mapsto \frac{az+b}{cz+d}$

Equivalently, the conformal automorphism group of the Riemann sphere is precisely the group of fractional linear transformations: namely, $PSL(2,\mathbb{C})$.

Proof

The proof involves the following steps:

• We first observe that the conformal automorphism group acts transitively on the Riemann sphere. In particular, any point $z_0\in \mathbb{C}$ can be mapped to $\mathrm{\infty }$ can be sent to $\mathrm{\infty }$ using the map:

$z\mapsto \frac{1}{z-z_0}$

• The isotropy group at $\mathrm{\infty }$ must comprise maps of the form $z\mapsto az+b$, because any conformal automorphism of the Riemann sphere that fixes $\mathrm{\infty }$ must be a conformal automorphism of $\mathbb{C}$, and any conformal automorphism of the complex numbers must be an affine map.