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As known, all finite subgroups of SO(3), except for Cyclic and Dihedral are isomorphic to $A_4, S_4$ or $S_5$. The classical proof of this fact uses geometry of Regular polyhedrons, their symmetries and rotations. Are there any algebraic proofs?(I mean any proof that takes SO(3) as a group of matrices or operators, but not as a group of ratios of three-dimensional space.

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