divisible group

An abelian group $D$ is said to be divisible if for any $x\in D$ , $n\in\Z^+$ , there exists an element $x'\in D$ such that $nx'=x$ .

Some noteworthy facts:

• An abelian group is injective (as a $\Z$ -module) if and only if it is divisible.
• Every group is isomorphic to a subgroup of a divisible group.
• Any divisible abelian group is isomorphic to the direct sum of its torsion subgroup and $n$ copies of the group of rationals (for some cardinal number $n$ ).