I am relatively new to isomorphisms but have been studying the subject and came up across this question:
Determine whether the groups $(\Bbb Z,+)$ and $ (\Bbb Q_{>0}, \cdot)$ are isomorphic to each other.
The answer to this question claims that these two groups are isomorphic but I believe this is false. Firstly, surely it must be impossible to have a non-cyclic group that is isomorphic to a cyclic one. And secondly, if $(\Bbb Z, +) \cong (\Bbb Q_{>0}, \cdot)$ then as $1$ generates $\Bbb Z$ then surely the image of $1$ over the isomorphism must generate $\Bbb Q_{>0}$, again which is impossible since this group has no generator.
Best Answer
You're quite right: if $G$ is cyclic (generated by $g$, say) and is isomorphic to $H$, with $\phi$ an isomorphism, then $\phi(g)$ generates $H$. (This is a good exercise for you.)