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I was wondering if it's possible to construct an isomorphism from the group of real numbers with addition to the group of nonzero real numbers with multiplication. It doesn't seem like it should be.

Jen.
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    There is, however, an isomorphism between $(\mathbb{R}, +)$ and $(\mathbb{R}^{>0}, \times)$ (namely, $x \mapsto e^x$). – Austin Mohr Feb 05 '14 at 01:55
  • How about between R×(Z/2Z) and (R∗,⋅)? What would that sort of isomorphism even look like? – Jen. Feb 05 '14 at 01:56
  • @Jen If you have another question, please ask it in a separate post, not in the comments for this one. – MJD Feb 05 '14 at 02:15
  • @Jen: Yes, $\mathbb{R}^*$ is isomorphic to $\mathbb{R}^+ \times \mathbb{Z}/2$. I think you can figure this out. – Martin Brandenburg Feb 05 '14 at 02:26

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The group $(\mathbb{R}\setminus\{0\},\times)$ has an element of order $2$, namely $-1$. However, $(\mathbb{R},+)$ does not have an element of order $2$ (since $a+a=0$ implies $a=0$).

Rebecca J. Stones
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