Verbal subgroup of abelian group implies divisibility-closed

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., verbal subgroup of abelian group) must also satisfy the second subgroup property (i.e., divisibility-closed subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about verbal subgroup of abelian group|Get more facts about divisibility-closed subgroup

Statement

Suppose $G$ is an abelian group and $H$ is a verbal subgroup of $G$ (so $H$ is a verbal subgroup of abelian group). Then, $H$ is a divisibility-closed subgroup of $G$. In other words, for any prime number $p$ such that $G$ is $p$-divisible, $H$ is also $p$-divisible.

Facts used

1. Verbal subgroup equals power subgroup in abelian group

Proof

The proof follows from Fact (1) and a little more work. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.