Finite subgroup of GL_2(C). Can anyone tell me a good reference(preferably book) to know all the finite subgroup of $GL_2(C)$,where $C$ is the field of complex numbers.I need it as I want to study their ring of invariance under the action on $C[X,Y]$.
with best, 
anjan  
 A: Will you settle for the finite subgroups of $SL_2(C)$?
$SU_2(C)\subset SL_2(C)$ is isomorphic to the unit quaternions.  The finite subgroups of the quaternions are not hard to classify; Google turns up this reference, for example.  Every finite subgroup of $SL_2(C)$ is conjugate to one of these.  
A: The list is presented in [G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274–304]. 
It is also in [P. Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964], and I am told that also on Coxeter's Regular Complex Polytopes, but I have not been lucky enough to have that book in my hands.
(As for the rings of invariants, I am pretty sure they have been explicitly given in various places. For the special unitary finite groups, they are in Klein's Lectures on the icosahedron, and the non-unitary ones should not be hard to produce from those because the "special subgroup" of these groups is normal with cyclic quotient)
