It is well known that the finite subgroups of $SL(2,\mathbb{C})$ up to conjugacy are the binary polyhedral groups (or Klein groups). There are two infinite families (cyclic groups and binary dihedral groups) and three exceptional groups (binary tetrahedral, binary octahedral, binary icosahedral). These groups and the related quotient singularities have a lot of interesting structure and properties. Some of them are listed in the following old and very nice question:

The finite subgroups of SL(2,C)

I am interested in the following two related questions.

1) What about the finite subgroups of $GL(2,\mathbb{C})$? Every finite subgroup of $GL(2,\mathbb{C})$ can be obtained by extending a finite subgroup of $SL(2,\mathbb{C})$ with a cyclic group of appropriate order. However, if we do not pay attention we might obtain the same subgroup twice. Do you know a reference where this is worked-out carefully and a list of the groups together with generators for each group are given explicitely?

2) What about the finite subgroups of $SL(2,\mathbb{k})$ and $GL(2,\mathbb{k})$ where $\mathbb{k}$ is an algebraically closed field? I am interested in finite groups $G$ such that the characteristic of $\mathbb{k}$ does not divide $|G|$. Can we say that we have exactly the same groups as in the complex situation? Again any reference in this direction is highly appreciated. Since most references I found work only over $\mathbb{C}$, I wonder where problems may arise.

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    $\begingroup$ A complete answer is here I believe, along with much more: math.unice.fr/~beauvill/pubs/PGL%282%29.pdf (Essentially you get the same groups as in the complex case.) $\endgroup$ – Nick Gill Dec 3 '19 at 16:05
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    $\begingroup$ Note also that if $X$ is a finite subgroup of ${\rm GL}(2,\mathbb{C})$ we can multiply each generator $x$ of $G$ by a scalar matrix $z$ so that $xz$ has determinant $1$. The subgroup $Y$ of ${\rm SL}(2,\mathbb{C})$ generated by such $xz$'s satisfies $Y/Z(Y) \cong X/Z(X)$. The same argument works with any algebraically closed field of characteristic coprime to $|G|.$ $\endgroup$ – Geoff Robinson Dec 3 '19 at 21:31
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    $\begingroup$ I think that finite subgroups of $GL_2(D)$ for $D$ a division ring have been classified by B. Hartley and his students around 1980. My supervisor M. A. Shahabi was his student and as I know, his thesis was on this subject, and some other students of Hartley also have similar works. You may check MathRev or ZB. Also there is a book by Shirvani. $\endgroup$ – M. Shahryari Dec 4 '19 at 7:18
  • $\begingroup$ @GeoffRobinson Thank you! It is indeed a nice argument to reduce to the case $G\subset SL(2,\mathbb{k})$. $\endgroup$ – Alessio Dec 6 '19 at 13:28
  • $\begingroup$ @M.Shahryari Thank you! Although I am mainly interested to the field situation, I would be happy to have a look at the division ring case as well. I couldn't find Shahabi's Ph.D. thesis unfortunately. $\endgroup$ – Alessio Dec 6 '19 at 13:31

An answer to your questions is provided by this article of Beauville:

Beauville, Arnaud, Finite subgroups of (\mathrm{PGL}_2(K))., García-Prada, Oscar (ed.) et al., Vector bundles and complex geometry. Conference on vector bundles in honor of S. Ramanan on the occasion of his 70th birthday, Madrid, Spain, June 16–20, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4750-3/pbk). Contemporary Mathematics 522, 23-29 (2010). ZBL1218.20030.

A final version is available here.

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  • $\begingroup$ Thank you Nick! Beauville's paper completely answers my question about what happens over arbitrary fields. It is really nice! However, the classification over $\mathbb{C}$ is much older of course. I am still interested to find (for historical reasons) an older reference for that. Do you know one? $\endgroup$ – Alessio Dec 6 '19 at 13:35
  • $\begingroup$ Sorry Alessio, I'm not sure what the best reference would be. Maybe the original result is due to Klein? But that's just a wild guess... $\endgroup$ – Nick Gill Dec 6 '19 at 13:44
  • $\begingroup$ I also suspect that the original result should be due to Klein, but couldn't find a precise reference yet. Thank you anyway! $\endgroup$ – Alessio Dec 6 '19 at 13:48

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