Show that the group $SO_3$ has infinitely many non-Abelian subgroups

Question

I have just shown that $SO_3$ acts transitively on the unit sphere, and that there is an order-$n$ subgroup for all positive integers $n$.

I'm looking for a hint with where to start to find the construction of infinitely many non-Abelian subgroups?

Answer 1

You can argue by cardinality, as $SO(3)$ is not enumerable nor abelian. One can construct by induction a strictly increasing sequence $G_n<G_{n+1}$ of non abelian subgroups.

For $G_1$, consider the subgroup generated by 2 non commuting rotations (eg two rotation of angle $\pi \over 100$ with different axis), so that they do not commute. Assume $G_n$ is constructed. As it is enumerable, there exists a rotation $g_{n}\notin G_n$. Let $G_{n+1}$ the subgroup generated by $g_n$ and $G_n$.