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I was reading $\textit{A Report on Artin's Holomorphy Conjecture}$ by Dipendra Prasad and C. S. Yogananda. (http://www.math.tifr.res.in/~dprasad/artin.pdf)

On p. 9, they state that the finite subgroups of $\textrm{GL}_2(\mathbb{C})$ can be classified according to their images in $\textrm{PGL}_2(\mathbb{C})$, being one of the following:

1) Cyclic,

2) Dihedral,

3) Tetrahedral,

4) Octahedral,

5) Icosahedral.

They attribute the classification of the finite subgroups of $\textrm{GL}_2(\mathbb{C})$ to Felix Klein, but they do not provide a reference. After extensive research, I have been unable to find a resource that treats the matter.

This is of course what the progress made on the proof of Artin's conjecture in the case of two-dimensional representations hinges upon.

If anyone could provide a reference, or briefly explain why the above list is an exhaustive classification of the finite subgroups of $\textrm{GL}_2(\mathbb{C})$, I would be most grateful.

$\textbf{Addendum}:$ I have still not solved the above and therefore add a bounty. I add that on p. 25 of $\textit{Base Change for}\ \textrm{GL}(2)$ by R.P. Langlands, he mentions in passing that $\textrm{PGL}(2,\mathbb{C}) \cong \textrm{SO}(3,\mathbb{C})$, which is significant. But it still remains to show that the finite subgroups of $\textrm{SO}(3,\mathbb{C})$ fall into one of the five classes listed above, which is not obvious to me. The list is of course reminiscent of the five known platonic solids: The tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. But if there is a relationship, I fail to see it.

Heinrich Wagner
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    Finite subgroups of $PSL(2,C)$ are all isomorphic to finite subgroups of $SO(3)$ and the latter are known since Antiquity (see your list). The story with subgroups of $GL(2,C)$ is more complicated since it involves central extensions of the groups on your list. Are you use the authors did not mean $SL(2,C)$? – Moishe Kohan Dec 21 '19 at 22:20
  • If you search for "finite subgroups of gl(2,c)" then you will find lots of references, including previous questions to this forum such as this one which was closed. – Derek Holt Dec 22 '19 at 11:31
  • @MoisheKohan I commend you for very elegantly explaining why the finite subgroups of $\textrm{PSL}(2,\mathbb{C})$ can be classified in the above manner. The aim is to classify all two-dimensional representations of finite groups, so the object under consideration must be $\textrm{GL}(2,\mathbb{C})$, as we also want to include matrices whose determinants are non-trivial roots of unity. – Heinrich Wagner Dec 22 '19 at 14:47
  • @DerekHolt I did search for that but was unable to find any reference worth its salt. I did also find the question you are referring to, but as you mentioned: The question was closed - not, it appears, due to the irrelevance of the question, but because it was poorly formulated; and so the question received no satisfactory answer. – Heinrich Wagner Dec 22 '19 at 14:48

2 Answers2

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The following paper https://www.researchgate.net/publication/254410139_Algebraic_subgroups_of_GL_2C

provides all algebraic subgroups of $\mathrm{GL}_2(\mathbb{C})$ in Section 2, but you certainly can extract the list of all finite subgroups. You should read first the considerations at the very beginning of Section 2 and then go to Theorem 4.

GreginGre
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  • I had a look at that paper. Their theorem 2 is exactly what I need. But they don't prove it. They only make reference to the book by Klein, which can be found by following this link: https://archive.org/details/cu31924059413439/page/n17. They do not even specify where in the book this proof is to be found. – Heinrich Wagner Dec 31 '19 at 17:33
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One can start with the classification of the finite subgroups of $SL_2(\Bbb C)$ and the unit quaternions $SU(2)$ inside $SL_2(\Bbb C)$ (any finite subgroup $G$ of $SL_2(\Bbb C)$ can be made to preserve an Hermitian inner product on $\Bbb C^2$ by averaging, hence is also a finite subgroup of $SU(2)$). This case is well documented. For several references see this MO-question, e.g., by Milnor, Slodowy and Dimca. In fact Dolgachev gives a nice and detailed proof here in section $1.1$.

The next step is extending this to $GL_2(\Bbb C)$. But again, this is well-documented at this MO-question, even for other fields $K$, e.g., for $GL_2(K)$.

Dietrich Burde
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