Is $(\mathbb{Q} \times \mathbb{Q}, +)$ isomorphic to the group $(\mathbb{Q}, +)$?
I have already found that neither groups are cyclic, however am unsure how to prove or disprove an isomorphism. I know that there needs to be a bijection and a homomorphism, however my course has not yet covered homomorphisms so I don't understand how to use them in this situation. I also know that for the group to be isomorphic it has to be commutative – could I use this somehow?
Best Answer
Hint: If $f:\mathbb Q \to \mathbb Q \times \mathbb Q$ is additive, then $f(q)=qf(1)$ and so $f$ cannot be surjective.