# 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 $nx^{\prime}=x$.

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 DivisibleGroup 2013-03-22 13:47:17 2013-03-22 13:47:17 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 20K99