divisible group

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

Some noteworthy facts:

• An abelian group is injective (http://planetmath.org/InjectiveModule) (as a $\mathbb{Z}$-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 $n$).

Title	divisible group
Canonical name	DivisibleGroup
Date of creation	2013-03-22 13:47:17
Last modified on	2013-03-22 13:47:17
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Definition
Classification	msc 20K99