show that in the category of divisible abelian groups, natural mapping $\pi:\mathbb{Q} \rightarrow \frac{\mathbb{Q}}{\mathbb{Z}}$ is monic but not one to one.
if you give me hint,Idea or reference that will be great,my first time studying homology algebra and it is self study.thanks a lot.
If $A$ is divisible and $A \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$ vanishes, then $A \to \mathbb{Q}$ vanishes. In fact, $A \to \mathbb{Q}$ factors through $A \to \mathbb{Z}$ by assumption, so that the image is a divisible subgroup of $\mathbb{Z}$. The only divisible subgroup of $\mathbb{Z}$ is trivial.
This shows that $\mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$ is a monomorphism in the category of divisible abelian groups. Of course, it is not injective, since the kernel in the category of abelian groups is $\mathbb{Z}$. (The kernel in the category of divisible abelian groups is zero!)