I'm studying the book Rick Miranda, Algebraic Curves and Riemann Surfaces and I have a question about the exercise H of page 84.
The book says that $z \mapsto exp(2\pi i /r)z$ is an automorphism of the Riemann Sphere $\mathbb{C}_{\infty}$ but it's not. It's not even injective. I don't know how to interpret this. I know that I can prove that the group generated by this two elements it's a dihedral group. But I'm interested in this example because I need actions over riemann surfaces (specially from dihedral groups).
EDIT: Let $ e^{2\pi i /r} = \zeta$. I misread because the map was $ z \mapsto \zeta \cdot z$ (Thanks Daniel Fischer).
The real question:
I want to prove that there are three branch points to the quotient map, with ramification numbers $2,2,r$. This is the same as proving that there exist $3$ orbits of size $r,r,2$.
I found one orbit of size $r$. It's the set of $r$-roots of unity.
I don't know how to find the other two orbits.