Let be a nonzero integer. An abelian group is called -divisible if for every , there exists such that . An abelian group is called n-torsionfree if for some implies . An abelian group is called uniquely -divisible if for any , there exists exactly one such that .

Let be the map

(a) Prove that is -torsionfree iff is injective and that is -divisible iff is surjective.

(b) Now suppose is an exact sequence of abelian groups. It is easy to check that the following diagram commutes:

Suppose that is uniquely -divisible. Prove that is -torsionfree if and only if is -divisible.