Exercise 1.6.6 (The additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic)

Proof. Assume for the sake of contradiction that there is an isomorphism $\phi :\mathbb{Q}\to \mathbb{Z}$. Put $a={\phi }^{-1}(1)\in \mathbb{Q}$ and $x=a∕2$ so that $x+x=a$.

Since $\phi$ is a homomorphism,

so $2\phi (x)=1$, where $\phi (x)\in \mathbb{Z}$, thus $2\mid 1$. But $2\nmid 1$: this contradiction shows that there is no isomorphism $\mathbb{Q}\to \mathbb{Z}$. □