Is $\Bbb C^* \times \Bbb Z$ isomorphic to a subgroup of $\Bbb C^*$?

View $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Since $\dim_{\mathbb{Q}} \mathbb{R} = \mathfrak{c}$, it follows that there is a $\mathbb{Q}$-vector space isomorphism $\mathbb{R}\times \mathbb{Q} \to \mathbb{R}$. Thus we have an injective group homomorphism $\mathbb{R}\times \mathbb{Z} \hookrightarrow \mathbb{R}$. This induces an injective group homomorphism

$$S^1 \times \mathbb{R} \times \mathbb{Z} \hookrightarrow S^1 \times \mathbb{R}.$$