Finitely generated torsion subgroup of $SO(3,\mathbb{R})$ is finite

Question

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.

Answer 1

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.