The factor group $\mathbb{R}^{*}/\{-1,1\}$ is isomorphic to $\mathbb{Z}_{2}$. True or False. I have been told that the answer for this question is False. And I'm trying to understand why.
For what I understand so far, $\mathbb{Z} _2$ is an abelian group. I also understand that for a group isomorphism of two groups, say: 
$\phi: G\rightarrow H$ 
*If $G$ is abelian, then $H$ is abelian.
*If $G$ is cyclic, then $H$ is cyclic.
*If $G$ has a subgroup of order $n$, then $H$ has a subgroup of order $n$.
But I'm not entirely sure if it might be related to the question or not. However, any hint would be really helpful. Thank you. 

Hint: $\mathbb{Z}_2 $ is finite, while $\mathbb{R}^*/\{-1,1\}$ is infinite.
