Verbality is strongly join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., verbal subgroup) satisfying a subgroup metaproperty (i.e., strongly join-closed subgroup property)
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Statement

Suppose $G$ is a group and $H_{i},i\in I$ are verbal subgroups of $G$. Then, the join of subgroups $\langle H_{i}\rangle$ is also a verbal subgroup of $G$.

Related facts

• Verbality is transitive
• Verbality is quotient-transitive
• Verbality satisfies image condition

Proof

We simply look at the sets of words for each $H_{i}$, and take the set of products of all such words.