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Does $SO(3)$ have an open nontrivial subgroup?(Group $SO(3)$ with usual matrices product, is all $3\times 3$ matrices whose determinant is 1 and for every element $A\in SO(3)$ we have $A^tA=AA^t=I_3$ and also let it's norm be a operator norm, that's, norm of linear mapping $A:R^3\rightarrow R^3$ which induced the topology on $SO(3)$.)

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Note that open subgroups are also closed (since all cosets are open, any coset, being the complement of the union of the other cosets, is also closed). So we'd be talking about a subgroup that is open and closed. But $SO(3)$ is connected; in fact it is a quotient space $S^3/\{1, -1\}$ of the unit quaternions, which is itself connected. Thus there are no nontrivial open subgroups.

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