Divisible abelian group

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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