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Is there an easy way to prove that every subgroup of $SU(2)$ is closed? If there isn't, is there a reference for this fact?

I thought that this was left as an exercise on the book on compact Lie groups by Sepanski, but I cannot find the reference. Thanks.

José Carlos Santos
  • 427,504
Max Reinhold Jahnke
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1 Answers1

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No. Take, for instance, the group of those matrices of the form$$\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix},$$with $\theta\in\mathbb Q$.

José Carlos Santos
  • 427,504
  • I just realized I asked the wrong question. The subgroup in question is actually the integral group of a subalgebra of $\mathfrak{su}(2)$. That is, is it true that any $K = \exp_{SU(2)}(k)$ is closed for every subalgebra $k \subset \mathfrak{su}(2)$? Should I accept your answer here or should I ask another question? – Max Reinhold Jahnke Jul 10 '18 at 23:22
  • @MaxReinholdJahnke You should do both. – José Carlos Santos Jul 11 '18 at 06:18
  • Thank you very much. I just accepted your answer and posted the other question here: https://math.stackexchange.com/questions/2847563/is-every-lie-subgroup-of-su2-closed – Max Reinhold Jahnke Jul 11 '18 at 11:25