Divisible group

Summary

A divisible group is an abelian group in which every element can be divided by every positive integer. The rational numbers, the real numbers and $\Q/\Z$ are divisible; the integers are not. Divisible groups are exactly the injective $\Z$-modules, and every divisible group is a direct sum of copies of $\Q$ and of Prüfer groups. Every abelian group embeds in a divisible group, and a divisible subgroup is always a direct summand.