# Are the groups $(\mathbb{R} \!\,, +)$ and $(\mathbb{R} \!\,^*,\dot{\,} \!\,)$ isomorphic?

I have an exercise to do and I don't understand how to solve it. It states:

Determine if the groups $(\mathbb{R} \!\,, +)$ and $(\mathbb{R} \!\,^*,\cdot{\,} \!\,)$ are isomorphic and to justify the answer.

I am not very good at this, but I would like a full demonstration.

$\mathbb{R}^* = \mathbb{R}\backslash\{0\}$

Thank you very much!

• Have you thought about torsion elements? – anon Nov 27 '13 at 22:21

$\textbf{Hint:} \:\:(-1)^2 = 1^2$
Regarding neat comment of @anon, note that: $$t(\mathbb R,+)=\{0\}\neq\{\pm1\}=t(\mathbb R^*,\cdot)$$ So you have different elements in both group which satisfy $x^n=id|_G$ where $n\in\mathbb N$.