Finite subgroups of $PGL_2(\mathbb{C})$ which have an orbit of size 2 on $\mathbb{P}^1_\mathbb{C}$ Is it true that any finite subgroup of automorphisms of $\mathbb{P}^1_\mathbb{C}$ which has an orbit of size 2 is either cyclic or dihedral?
Does anyone have a reference for this?
In general is it possible to describe the finite order automorphisms of $\mathbb{P}^1_\mathbb{C}$?
 A: I think yes. We have a morphism $SL_2(\mathbb C) \to PGL_2(\mathbb C)$ of index $2$, and the statement is true for $SL_2$. It is true for $SL_2$ because its finite subgroups are conjugated to subgroups of $SU_2(\mathbb C)$ and they form a double cover of $SO(3)$. In fact, any finite group acting on the sphere will correspond to a regular solids and there is orbit of cardinal two only for "degenerate" regular solids (corresponding to cyclic and dihedral group). All of this is proven in the book of Lamotke, Regular solids and isolated singularities.
A: The linear fractional mappings that fix infinity form a group isomorphic to group of affine transformations $x\to ax+b$, where $a\ne0$. The subgroup of this that fixes 0 consists of the maps $x\to ax$. Since $PGL(2,\mathbb{F})$ is 2-transitive (actually 3-transitive) on the line, it follows that any subgroup of $PGL(2,\mathbb{F})$ that fixes two points is isomorphic to a subgroup of the multiplicative group of our field.
If a subgroup $G$ has an orbit of length two, it has a normal subgroup of index two with two fixed points, and so this subgroup is cyclic. An element of $PGL$ that swaps $0$ and $\infty$ must have the form $x\to b/(cx)$, given this it follows that $G$ is cyclic or dihedral.
