Proving Two Groups are not Isomorphic

Question

How do I show that the reals under addition are not isomorphic to the rationals under addition, and that the rationals under addition are not isomorphic to the integers under addition?

I've tried finding elements with different orders, but for each group, the elements have infinite order. I know all three groups are Abelian. Is it because they won't form a bijection?

Answer 1

There is no bijection from the reals to neither the rationals nor to the integers. Specifically, there can be no isomorphisms.

As for the integers vs. the rationals, say there is an isomorphism $\phi:\Bbb Q\to \Bbb Z$. Let $p\in \Bbb Q$ be such that $\phi(p)=1$. What property does $\phi(p/2)\in \Bbb Z$ have? Is there such an integer?

Answer 2

Since $\mathbb{Z}$ is cyclic with generator $1$, you can look at the image of $1$ under your supposed isomorphism. No matter what it is, it can't generate $\mathbb{Q}$.