Given that there is the additive group $\mathbb Q$ of rational numbers, and the multiplicative group $\mathbb Q^*$ of non-zero rational numbers, prove that $(\mathbb Q,+)$ is not isomorphic to $(\mathbb Q^*,\times)$.
How many methods can you think of and can you provide a complete solution?
I am a self-learner of maths and feel difficult to offer a rigorous proof, but here are my thoughts:
I could try to assume a isomorphism $\theta$ exists between the two groups and prove that $\theta$ cannot exist.
I could try to find some property which should preserve under isomorphism but is satisfied only by one of the groups.
However I could not proceed in either direction, could someone please help?