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 $\mathbb {C} _{\infty }$ (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 $\pm 1$.