Verbality is finite direct power-closed

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

Suppose $H$ is a verbal subgroup of a group $G$. Let $n$ be a natural number. Then, in the $n^{th}$ direct power $G^{n}$ of $G$ (i.e., the external direct product of $G$ with itself $n$ times) the corresponding subgroup $H^{n}$ is a verbal subgroup. In fact, the same set of words works.

Related facts

• Full invariance is finite direct power-closed
• Homomorph-containment is finite direct power-closed

Proof

Given: A group $G$, a collection $C$ of words, $H$ is the subgroup of $G$ comprising those elements of $G$ that can be expressed as a product of elements that can be expressed in $G$ as words from $C$. A positive integer $n$. An element $h=(h_{1},h_{2},\dots ,h_{n})\in H^{n}$.

To prove: $(h_{1},h_{2},\dots ,h_{n})$ is in the verbal subgroup of $G^{n}$ corresponding to the collection$C$.

Proof: We know that there exist words $w_{1},w_{2},\dots ,w_{n}$ such that each $w_{i}$ is expressible as a product of words from $C$, and elements $g_{1,1},\dots ,g_{1,m_{1}},g_{2,1},\dots ,g_{2,m_{2}},\dots ,g_{n,1},\dots ,g_{n,m_{n}}$ such that:

$h_{i}=w_{i}(g_{i,1},\dots ,g_{i,m_{i}})$

Consider $w$ as the word that is the product of the $w_{i}$s, with different input letters for each $i$. Then, $w$ is also a word generated by $C$, and in fact $h$ is in the image of the word map corresponding to $w$, completing the proof.