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I am aware of a theorem stating that a finitely generated torsion subgroup of $GL_n(\mathbb{C})$ is finite. I am trying to prove a more humble version of the theorem, namely that a finitely generated torsion subgroup of $SO(3,\mathbb{R})$ is finite and am really stuck. Any suggestions will be deeply appreciated.

yoni
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The Theorem you are mentioning is due to Schur, in the context of the Burnside Theorem:

Theorem (Schur, 1911): Every finitely generated periodic subgroup of $GL(n,\mathbb{C})$ is finite.

Here periodic means torsion. Now for $SL_3(\mathbb{R})$ the finite subgroups are classified here, namely $C_n,D_n,A_4, S_4,A_5$. I suppose this can be used to see that they exhaust all finitely-generated torsion subgroups.

Edit: The reference is Schur I., Über Gruppen periodischer Substitutionen, Sitzber. Preuss. Akad. Wiss. (1911), 619–627.

Dietrich Burde
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  • This also follows from Selberg's Lemma, which has a fairly easy proof here: https://www.researchgate.net/publication/240458734_An_elementary_account_of_Selberg%27%27%27%27s_Lemma – Steve D Dec 05 '17 at 20:47
  • @SteveD: Yes, except it was proven about 60 years before Selberg. (I think, I even saw it in Burnside's book "Theory of groups of finite order", 1897.) – Moishe Kohan Dec 05 '17 at 20:50
  • @MoisheCohen: right, the point is that the link I posted might be a slightly easier reference, to arrive at the same end goal. – Steve D Dec 06 '17 at 00:40