The setting is the extended complex plane which includes

Question

Question: 1. The setting is the extended complex plane, which includes the point at infinity.

A point z₀ is a fixed point of a mapping f if f(z₀) = z₀ Suppose f is a linear fractional transformation that is neither a translation nor the identity mapping f(z) = z. Prove that f must have either one or two fixed points, but cannot have three. Why does this conclusion fail to hold for translations?

How many fixed points can a translation have?

2. The setting is the extended complex plane, which includes the point at infinity.

Let f be a linear fractional transformation with three fixed points. Prove that f is the identity mapping.