Non exact sequence of quotients by torsion subgroups $$0\rightarrow G_{1}\rightarrow G_{2}\rightarrow G_{3}\rightarrow 0$$ is a short exact sequence of finitely generated abelian groups. We call $\bar{G_{i}}$ the quotient of $G_{i}$ by its torsion subgroup.
I want to show that $$0\rightarrow \bar{G_{1}}\rightarrow \bar{G_{2}}\rightarrow \bar{G_{3}}\rightarrow 0$$ is not exact. I am looking for example with $\mathbb{Z}/n\mathbb{Z}$ or something else... Can you help me ? Thank you.

Suppose your sequence is always exact. Then take $G_2$ any torsionfree group and $G_1$ any subgroup of $G_1$. Then also $G_1$ is torsionfree and exactness would imply that $G_2/G_1$ is torsionfree.
But for every abelian group $G$ there exists a surjection $F\to G$, with $F$ a free group (in particular torsionfree). So every group would be torsionfree.
