Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$?

Question

Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$?

Answer 1

No. $(\Bbb Z, +)$ is generated by $\{1,-1\}$ and $(\Bbb Q^\times, \cdot)$ is not finitely generated.

Answer 2

They are not isomorphic. In $(\mathbb Q\setminus \{0\},\cdot)$ you have an element that is its own inverse $(-1)$. This does not happen in $(\mathbb Z,+)$