Show that ${\mathbb {R^\times} } \cong C_2 \times (\mathbb{R},+)$

Question

Show that ${\mathbb{R^\times}} \cong C_2 \times (\mathbb{R},+)$.

I know that the group $C_2$ is the cyclic group of order 2, and that $(\mathbb{R},+)$ is the group of reals under addition. However, I'm stumped on how to construct an isomorphism between the reals under multiplication and the the cyclic group times the reals under addition. Anyone who can give a rigorous argument would be highly appreciated.

Answer 1

Hint:

Can you see how to construct an isomorphism between the group of POSITIVE reals under multiplication and $(\mathbb{R},+)$?

Answer 2

This is mainly a pattern recognition exercise. If you write out the definitions, among the things you are being asked to do is to find an inverse pair of functions $f$ and $g$ such that

• $f(xy) = f(x) + f(y)$
• $g(x+y) = g(x) g(y)$

You're expected to realize that you're familiar with two specific functions having these properties. When you try to build the isomorphism using them, the exercise follows rather straightforwardly.