Group of fractional linear transformations

Definition

The group of fractional linear transformations is defined in many equivalent ways:

• It is the conformal automorphism group of the Riemann sphere: i.e. it is the group of all biholomorphic mappings from the Riemann sphere to itself, under composition
• It is the group of maps from ${\mathbb{C}}_{\mathrm{\infty }}$ (the Riemann sphere) to itself, that can be expressed in the form:

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

• It is the group of those maps from $\mathbb{C}{\mathbb{P}}^1$ to itself that are induced by linear automorphisms of ${\mathbb{C}}^2$
• It is the group $PSL(2,\mathbb{C})$: the quotient of the group $SL(2,\mathbb{C})$ of matrices with determinant one, by the subgroup comprising the scalar matrices $\pm 1$.