$G=(\mathbb{R}^*,\cdot)$ , $N=\{-1,1\}$. prove that $G/N\cong (\mathbb{R}_+,\cdot)$

Question

Let $G=(\mathbb{R}^*,\cdot)$ and $N=\{-1,1\}$. I need to prove that $G/N\cong (\mathbb{R}_+,\cdot)$

This is all the details that I have in the exercise, usually given $\varphi$ such that $\varphi$ is a homomorphism, but here how colud I prove the isomorphism?

Answer 1

The absolute value is a map from the non-zero reals to the positive reals that conveniently is also a homomorphism with respect to multiplication.

Some further details need to be checked, which I do not spell out.