So I looked through the isomorphic properties and they seem to satisfy all of them...Prove that additive groups $\displaystyle \mathbb{Z}$ and $\displaystyle \mathbb{Q}$ are not isomorphic

- The order of $\displaystyle \mathbb{Z}$ equals the order of $\displaystyle \mathbb{Q}$

- Both are abelian

- The order of any element in the groups are all infinite (except the identity)

The only thing I can think of is that $\displaystyle \mathbb{Z}$ has a generator and $\displaystyle \mathbb{Q}$ doesn't, but does that have anything to do with them being isomorphisms?