the automorphisms of the plane and the sphere
Having understood the symmetries of the disk and the half-plane, it is natural to ask about the two other basic 'whole-space' domains: the entire complex plane C, and the Riemann sphere (the plane with a single point at infinity added). Each has its own complete list of conformal symmetries, and the answers are short, clean, and reveal a hierarchy: the bigger the domain, the more its symmetries are forced to behave.
For the whole plane C, the conformal automorphisms — holomorphic bijections of C onto itself — are exactly the non-constant affine maps f(z) = a z + b, with a not zero. That is all: a rotation-scaling by a followed by a translation by b. The proof leans on Liouville and the singularity at infinity: a self-map of the plane must have a pole (not an essential singularity) at infinity, which forces it to be a polynomial, and bijectivity forces that polynomial to have degree 1. For the Riemann sphere, the automorphisms are the full Mobius group: f(z) = (a z + b)/(c z + d) with a d - b c not zero, complex coefficients, modulo scaling — the group PSL(2,C). Every conformal symmetry of the sphere is a fractional linear transformation; there are no others. So the sphere's automorphism group is six-real-dimensional, the plane's is four-real-dimensional, the disk's and half-plane's are three.
This descending ladder is one of the most memorable facts in the subject. The sphere is the most symmetric (any three points can be sent to any three points by a unique Mobius map); the plane is less so (you can no longer move the point at infinity); the disk and half-plane are tightest of all (only three real parameters). An honest caveat that ties back to the Riemann mapping theorem: the WHOLE plane C is conformally unlike the disk — there is no conformal map from C onto the disk, by Liouville's theorem — which is exactly why C is excluded from the Riemann mapping theorem, and why C and D, though both simply connected, have genuinely different automorphism groups.
On the plane, f(z) = 2 z + 1 is an automorphism (scale by 2, shift by 1); f(z) = z^2 is NOT (it is two-to-one, not a bijection). On the sphere, f(z) = 1/z is an automorphism that swaps 0 and infinity — perfectly legal there because infinity is an honest point of the sphere, but impossible on the bare plane where infinity is missing.
The plane allows only affine maps; the sphere allows all Mobius maps, including z -> 1/z.
Do not confuse the plane and the sphere: z -> 1/z is an automorphism of the sphere but NOT of the plane (it is undefined at 0 and never attains the value 0 there). Adding the point at infinity is exactly what upgrades the affine group to the full Mobius group.