Is there any isomorphism from $\mathbb R^*$ to $\mathbb R^+ \times \mathbb Z_2$?

Question

Is there any isomorphism from $\mathbb R^*$ to $\mathbb R^+ \times \mathbb Z_2$?

I have tried so much but I fail to find such isomorphism,if any.Please help me.Thank you in advance.

Answer 1

Your questions is malformed. But to make sense of it, I will make the following assumptions:

$$\mathbb{R}^* \text{ and } \mathbb{R}^+ \text{ are being viewed as groups under multiplication.}$$

Then you can check that:

$$\varphi : \mathbb{R}^* \to \mathbb{R}^+ \times \mathbb{Z}_2$$

defined by $$ \varphi(r) =\begin{cases} (|r|, 0) & \text{ if } r > 0 \\ (|r|,1) & \text{ if } r < 0 \end{cases} $$ is a group isomorphism.

Here we are viewing $\mathbb{Z}_2$ as an additive group.