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)