Divisible abelian group

Definition
A divisible abelian group is an abelian group $$G$$ satisfying the following equivalent conditions:


 * 1) For every $$g \in G$$ and nonzero integer $$n$$, there exists $$h \in G$$ such that $$nh = g$$.
 * 2) Viewing the category of abelian groups as the category of modules over the rin of integers, $$G$$ is an injective module.

Examples

 * The group of rational numbers, and more generally, the additive group of any vector space over the field of rational numbers, is a divisible abelian group. In fact, it is a uniquely divisible abelian group.
 * The group of rational numbers modulo integers is a divisible abelian group.
 * The quasicyclic group for a prime $$p$$, i.e., the group of all $$(p^k)^{th}$$ roots of unity for all $$k$$ under multiplication, is also a divisible abelian group.

Facts

 * Divisible abelian group implies every automorphism is abelian-extensible
 * Divisible abelian group implies every endomorphism is abelian-extensible
 * Divisible abelian subgroup of abelian group contains no proper nontrivial verbal subgroup