Characteristic equals verbal in free abelian group

Statement
In a free Abelian group, any characteristic subgroup is a verbal subgroup. More explicitly, the only characteristic subgroups of a free Abelian group are the subgroups comprising the multiples of $$n$$ for particular choices of natural number $$n$$.

Related facts
The statement holds for a somewhat larger class of groups than free Abelian groups: for instance, it holds for all cardinality-restricted external direct products of the group of integers. In particular, it holds for the Baer-Specker group.

The statement also holds for cyclic groups and elementary Abelian groups -- however, it breaks down for other finite and finitely generated Abelian groups.

Some related facts are:


 * Characteristic equals fully characteristic in finite Abelian group
 * Characteristic not implies fully characteristic in finitely generated Abelian group
 * Fully characteristic not implies verbal in finite Abelian group