Abelian group is divisible if and only if its isomorphic to a direct sum of $\mathbb Q$ and quasicyclic groups I am looking for a reference for the fact that an Abelian group is divisible if and only if its isomorphic to a direct sum of $\mathbb Q$ and quasicyclic groups so I can study the proof. Can someone suggest me a reference?
 A: You should find it in Fuchs' “Infinite Abelian Groups” (volume 1).
However, the proof is not really so difficult. Here's a sketch.
Let $G$ be a divisible abelian group. Then the torsion part $t(G)$ is divisible as well and so it splits: $G\cong t(G)\oplus (G/t(G))$. Since $G/t(G)$ is torsionfree, it has unique division by integers and so it becomes a vector space over $\mathbb{Q}$, hence a direct sum of copies of $\mathbb{Q}$.
It remains to analyze $t(G)$, which can be written as the direct sum of the $p$-components, like every torsion group. Hence we are reduced to a torsion divisible $p$-group ($p$ a prime).
This is perhaps the most difficult part. But some module theory can help: since $\mathbb{Z}$ is noetherian, every injective module is a direct sum of indecomposable modules. Thus we just need to characterize the indecomposable divisible $p$-groups (injectivity on $\mathbb{Z}$-modules is the same as divisibility).
In an indecomposable divisible $p$-group the socle must be simple and so the group is the injective envelope of $\mathbb{Z}/p\mathbb{Z}$ which is easily seen to be the Prüfer $p$-group.
A: The structure theorem of divisible groups states that any divisible group is a direct sum of copies of the additive rationals and quasicyclic groups.
For references see for example here.
