Abelian implies every subgroup is potentially characteristic

Statement
In an Abelian group, every subgroup is a potentially characteristic subgroup: it can be realized as a characteristic subgroup inside some bigger group.

Related facts

 * Finite implies every normal subgroup is potentially characteristic
 * Nilpotent implies every normal subgroup is potentially characteristic
 * Hypercentral implies every normal subgroup is potentially characteristic

Facts used

 * 1) uses::Central implies potentially characteristic
 * 2) uses::Abelian implies every subgroup is central

Proof
The proof follows directly by piecing together facts (1) and (2).