Subgroup of finite abelian group implies finite-abelian-pi-potentially verbal

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subgroup of finite abelian group) must also satisfy the second subgroup property (i.e., abelian-potentially verbal subgroup)
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Statement

Suppose $G$ is a finite abelian group and $H$ is a subgroup of $G$. Then, there exists a finite abelian group $K$ containing $G$ and having no prime factors other than those of $G$, such that $H$ is a verbal subgroup of $K$.

In particular, every Subgroup of finite abelian group (?) is an Abelian-potentially verbal subgroup (?), Abelian-potentially fully invariant subgroup (?), and Abelian-potentially characteristic subgroup (?).

Related facts

Similar facts

• Central implies finite-pi-potentially verbal in finite
• Cyclic normal implies finite-pi-potentially verbal in finite
• Homocyclic normal implies finite-pi-potentially fully invariant in finite

Opposite facts

• Subgroup of abelian group not implies abelian-potentially characteristic, subgroup of abelian group not implies abelian-extensible automorphism-invariant

Proof

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