Epimorphism between free Abelian groups

Question

I have found a statement (without proof) that there is no epimorphism in Group category $\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}\times \mathbb{Z}\times \mathbb{Z} $ existing. Could you please help, why it's true?

Answer 1

Assume that there is an epimorphism $f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}$, then by quotient $\mathbb{Z\times\mathbb{Z}}\times\mathbb{Z}$ with $2\mathbb{Z}\times2\mathbb{Z}\times2\mathbb{Z}$, we also get an epimorphism $\hat{f}:\mathbb{Z}\times\mathbb{Z}\to(\mathbb{Z}/2\mathbb{Z})^{\oplus3}$. Since $(\mathbb{Z}/2\mathbb{Z})^{\oplus3}$ is 2-torsion, $\hat{f}$ factors as $\hat{f}:\mathbb{Z}\times\mathbb{Z}\to(\mathbb{Z}/2\mathbb{Z})^{\oplus2}\to(\mathbb{Z}/2\mathbb{Z})^{\oplus3}$, in which each morphism is an epimorphism, this is impossible if you compute the order of the two finite groups (4 is less than 8).