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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Use \times not x for $\times$ in math mode. – Cameron Williams May 18 '20 at 14:06
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2No. $\mathbb{Z}_4/\mathbb{Z}_2 = \mathbb{Z}_2$, but $\mathbb{Z}_4 \not = \mathbb{Z}_2\oplus \mathbb{Z}_2$. – anomaly May 18 '20 at 14:08
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1$SO(2)\times\Bbb{Z}_2$ is an abelian group. $O(2)$ is not. – Jyrki Lahtonen May 18 '20 at 14:12
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Something along the lines that you are thinking is the Schur–Zassenhaus theorem. – Cameron Williams May 18 '20 at 14:17