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For any abelian group $G$, there is a torsion subgroup $TG=\{g\in G, ng=0 \textrm{ for some non-zero integer} n\}$.

Now, let $A\to B\to C$ be an exact sequence of abelian groups.

Is it true that $TA\to TB\to C$ is exact? (every maps are restriction of given maps)

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No. Consider the exact sequence $\mathbb Q \to \mathbb Q/\mathbb Z \to 0$. This restricts to $0 \to \mathbb Q/\mathbb Z \to 0,$ which is no longer exact.

Another (closely related) example is given by $\mathbb Z \to \mathbb Z/n\mathbb Z \to 0,$ for any $n > 1$.

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