Group homomorphism containing the trivial kernel only

Question

Is it right to say that this is a group homomorphism and it only contains the trivial kernel?

$$\Phi : (\mathbb{R}\setminus\{0\}, ×) \longrightarrow (\mathbb{R}\setminus\{0\}, ×) : x \mapsto |x|$$

Answer 1

It is a homomorphism. But the kernel is not trivial, since $1$ and $(-1)$ are both in the kernel.