I have the following question that I am confused by:
Show that the groups $(\mathbb{R}-\{0\},\cdot)$ and $(\mathbb{R},+)\times(\mathbb{Z}_{2},+)$ are isomorphic.
I have been able to show that $(\{1,-1\},\cdot)\simeq(\mathbb{Z}_{2},+)$ and that $(\{1,-1\}\cdot)\times((0,\infty),\cdot)\simeq (\mathbb{R}-\{0\},\cdot)$. How would I finish off this problem? Thanks for any suggestions!
hint: Let $\exp:(\mathbb R,+) \to (\mathbb R_{>0},\cdot)$ be the exponential map and note that $$\exp(a+b)=\exp(a) \cdot \exp(b).$$