TITLE: finite subgroups of $\text{PGL}(2,\mathbb{C})$
QUESTION [6 upvotes]: It is known (probably since Klein) that the finite subgroups of $\text{PGL}(2,\mathbb{C})$ are up to isomorphy exactly the following.

The cyclic groups $C_{n}$,
the dihedral groups $D_{n}$ (with $2n$ elements),
the tetrahedral group $A_{4}$,
the octahedral group $S_{4}$,
the icosahedral group $A_{5}$.

However, I'm not able to find a (modern) reference including a proof for this fact. Can someone help me out?

REPLY [2 votes]: Using geometry it is pretty easy to see that every finite subgroup of $PGL(2,\mathbb{C})$ is isomorphic to a finite subgroup of $SO(3,\mathbb{R})$, which is the group of isometries of the unit sphere. 
The geometric connection is that $PGL(2,\mathbb{C})$ is isomorphic to the group of orientation preserving isometries of hyperbolic 3-space $\mathbb{H}^3$, via the upper half space model. Under this isometry, every finite subgroup of $PGL(2,\mathbb{C})$ has a fixed point $P \in \mathbb{H}^3$, and therefore acts by orientation preserving isometries on the unit tangent sphere $T^1_P(\mathbb{H}^3)$. So by using any isometry between $T^1_P(\mathbb{H}^3)$ and the standard unit sphere in $\mathbb{R}^3$ one obtains an embedding into $SO(3,\mathbb{R})$.
So now the problem has been reduced to listing the isomorphism types of the finite subgroups of $SO(3,\mathbb{R})$. On this topic there is a ton of modern literature, including the wikipedia page on point groups in 3-dimensions.