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