Let $\displaystyle n$ be a nonzero integer. An abelian group $\displaystyle A$ is called $\displaystyle n$-divisible if for every $\displaystyle x \in A$, there exists $\displaystyle y \in A$ such that $\displaystyle x=ny$. An abelian group $\displaystyle A$ is called n-torsionfree if $\displaystyle nx=0$ for some $\displaystyle x \in A$ implies $\displaystyle x=0$. An abelian group $\displaystyle A$ is called uniquely $\displaystyle n$-divisible if for any $\displaystyle x \in A$, there exists exactly one $\displaystyle y \in A$ such that $\displaystyle x=ny$.

Let $\displaystyle \mu_n : A \rightarrow A$ be the map $\displaystyle \mu_n(a)=na$

(b) Now suppose $\displaystyle 0 \rightarrow A \overset{f}{\rightarrow} B \overset{g}{\rightarrow} C \rightarrow 0 $ is an exact sequence of abelian groups. It is easy to check that the following diagram commutes:

Suppose that $\displaystyle B$ is uniquely $\displaystyle n$-divisible. Prove that $\displaystyle C$ is $\displaystyle n$-torsionfree if and only if $\displaystyle A$ is $\displaystyle n$-divisible.