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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

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  • $\begingroup$ This is close to other questions discussed on MO such as mathoverflow.net/questions/16026 $\endgroup$ Feb 19, 2011 at 17:57

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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.

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  • $\begingroup$ There are many links in the thread linked to by Jim in a comment above to great treatments of the case of $SL$. $\endgroup$ Feb 19, 2011 at 18:14
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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)

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