Characteristic equals verbal in free abelian group

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a free abelian group. That is, it states that in a Free abelian group (?), every subgroup satisfying the first subgroup property (i.e., Characteristic subgroup (?)) must also satisfy the second subgroup property (i.e., Verbal subgroup (?)). In other words, every characteristic subgroup of free abelian group is a verbal subgroup of free abelian group.
View all subgroup property implications in free abelian groups $|$ View all subgroup property non-implications in free abelian groups $|$ View all subgroup property implications $|$ View all subgroup property non-implications

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 not implies fully invariant in finite abelian group
• Characteristic equals fully invariant in odd-order abelian group