What is the automorphism group for $\mathbb{C}\setminus \{0\}$? Show that it has two connected components, and that the connected component containing the identity is transitive.
What is the automorphism group for $\mathbb{C}\setminus \{0,1\}$? For each element of the automorphism group, find its order and the points in $\mathbb{C}\setminus \{0,1\}$ which are fixed under it.
By autmorphisms, I mean the 'set' (or in this group) of all biholomorphic functions from the space $X$ to itself ($X$ may be $\mathbb{C}^*$ or $\mathbb{C}\setminus \{0,1\}$). With this terminology, I would say that $Aut(\mathbb{D})$ is the set of all linear fractional transformations $\frac{az-b}{cz-d}$ with $ad-bc\neq 0$.
We know that $Aut(\mathbb{C})$ is the set of all (non-constant) linear transformations/ I think that $Aut(\mathbb{C}^*)$ should be the set of all those linear transformations which take 0 to 0 and hence must be of the form $az$ for $a\neq 0$.