Subgroups of Special Orthogonal Group

[Kalinka35]

Homework Statement

I would like to show that SO₃ does not contain any subgroups that are isomorphic to SO₂ X SO₂.

Homework Equations

I know that any finite subgroup of SO3 must be isomorphic to a cyclic group, a dihedral group, or the group of rotational symmetries of the tetrahedron, cube, or icosahedron.

The Attempt at a Solution

I think SO2 x SO2 can't be isomorphic to a cyclic group since there is no way it can have only one generator, but I'm not entirely sure how to prove this. As for the others, the criteria are more complex and the whole direct product is sort of throwing me.

Thanks.

 

[Billy Bob]

I know that any finite subgroup of SO3 must be isomorphic to a cyclic group, a dihedral group, or the group of rotational symmetries of the tetrahedron, cube, or icosahedron.

Can you find a finite subgroup of SO2 x SO2 that is not isomorphic to any of those?