# A property of abelian categories

My aim is to show that the category of free abelian groups not an abelian category.

I read that I could fix $n \in \mathbb{N} \setminus \left\lbrace 0,1 \right\rbrace$ and consider the "multiplication by $n$ " map $\mathbb{Z} \rightarrow \mathbb{Z}$ is a mono and an epi but not an iso.

Can anyone provide me with a hint as to why this is a contradiction?

Thanks!

The “multiplication by $2$” in a free abelian group is monic (obviously) and epic because the cokernel (computed in the category of abelian groups) is torsion, so there's no non zero morphism from it to any free abelian group. However it's not an isomorphism, because it has no inverse.
• @B.S. Somebody tells me there are also non abelian groups, but I'm quite dubious about this assertion. ;-) “All groups, apart for some unfortunate exceptions, are abelian”. – egreg Dec 31 '13 at 11:41