Math 311: Complex Analysis — Automorphism Groups Lecture

Aut(C) = {analytic bijections f : C −→ C}. Any automorphism of the plane must be conformal, for if f ′(z) = 0 for some z then f takes the value f(z) with multiplicity n > 1, and so by the Local Mapping Theorem it is n-to-1 near z, impossible since f is an automorphism. By a problem on the midterm, we know the form of such automorphisms: they are f(z) = az + b, a, b ∈ C, a 6= 0. This description of such functions one at a time loses track of the group structure. If f(z) = az + b and g(z) = a′z + b′ then (f ◦ g)(z) = aa′z + (ab′ + b), f−1(z) = a−1z − a−1b.