# $O(2)$ is isomorphic to $SO(2)\times\mathbb Z_2$ [duplicate]

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 $$SO(2)\times\mathbb Z_2$$

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