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Which finite groups are isomorphic to groups of meromorphic functions on the whole complex plane under composition?

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    $\begingroup$ Of meromorphic function on what? $\endgroup$ Commented Dec 11, 2021 at 18:37
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    $\begingroup$ To be in a group under composition, the meromorphic functions in question have to be injective. This implies they have to be rational, and in fact they have to be Mobius transformations. See e.g. here $\endgroup$ Commented Dec 11, 2021 at 19:57
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    $\begingroup$ I note that this question is getting close votes, but I like it just because the answer is so beautiful. $\endgroup$ Commented Dec 11, 2021 at 20:14
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    $\begingroup$ The have to be Mobius transformations, so they have to be rotations of the Riemann sphere, so they are the finite groups known since Plato. $\endgroup$ Commented Dec 14, 2021 at 12:20

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