Let $H$ be the subgroup {-1,1} in $G=\mathbb{R}^{*}$. Is $G/H$ isomorphic to $G$?
I know the answer is "no," but I am not really sure why. Is it because $a$ and $-a$ generate the same coset? If that's the case, if we just looked at the positive reals, would we then get an isomorphism?
Thanks in advance