I need a proof that every abelian group is a subgroup of divisible group (to make sure that every object of the category of $\mathbb Z$-modules has injective resolution).
I found a proof on group props but it uses wreath product which is out of my range at that point in my life.
Can you show me a simpler proof?
Let $G$ be a group. It can be written as the quotient of a free group, say $G=F/N$. If $G$ is abelian, the commutator subgroup $F'$ is included in $N$, so $G$ is a quotient of the abelianization $F^{ab} \simeq \bigoplus\limits_{i \in I} \mathbb{Z}$, say $F^{ab}/M$. Clearly, $F^{ab}$ can be embedded into $Q=\bigoplus\limits_{i \in I} \mathbb{Q}$, so $G=F^{ab}/M$ is embedded into $Q/M$. But $Q/M$ is divisible as the quotient of a divisible group.
Edit: The source is this post.
