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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!

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hint: Let $\exp:(\mathbb R,+) \to (\mathbb R_{>0},\cdot)$ be the exponential map and note that $$\exp(a+b)=\exp(a) \cdot \exp(b).$$

  • 0
    How exactly does one include the set $(\mathbb{R},+)\times(\mathbb{Z}_{2},+)$? – 2017-02-20
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    Try to use $-\exp$. What you've shown is that $\mathbb R \times \mathbb Z_{2}$ is most essentially two copies of $(\mathbb R,+)$. What would be the most natural thing to do? – 2017-02-20
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    I'm not sure exactly. Is there some use for the first isomorphism theorem here? – 2017-02-20