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If have proven that $\frac{O(2)}{SO(2)}$ is isomporphic to $\mathbb Z_2$.

Does is follow from this that $O(2)$ is isomorphic to $SO(2)\times\mathbb Z_2$

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  • $\begingroup$ Use \times not x for $\times$ in math mode. $\endgroup$ May 18 '20 at 14:06
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    $\begingroup$ No. $\mathbb{Z}_4/\mathbb{Z}_2 = \mathbb{Z}_2$, but $\mathbb{Z}_4 \not = \mathbb{Z}_2\oplus \mathbb{Z}_2$. $\endgroup$
    – anomaly
    May 18 '20 at 14:08
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    $\begingroup$ $SO(2)\times\Bbb{Z}_2$ is an abelian group. $O(2)$ is not. $\endgroup$ May 18 '20 at 14:12
  • $\begingroup$ Something along the lines that you are thinking is the Schur–Zassenhaus theorem. $\endgroup$ May 18 '20 at 14:17

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