Verbality is transitive

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

Suppose $H\le K\le G$ are groups with each verbal in the next (i.e., $G$ is a group, $K$ is a verbal subgroup of $G$, and $H$ is a verbal subgroup of $K$). Then, $H$ is a verbal subgroup of $G$.

Related facts

• Verbality is quotient-transitive
• Verbality is strongly join-closed
• Verbality satisfies image condition

Proof

Proof idea

The idea is to "compose" the words by substituting. Explicitly, any element of $H$ can be written as a word of a certain type in terms of elements of $K$, and each of those elements of $K$ can be written as words of certain types in the element of $G$. We plug in those word expressions. Explicitly, if:

$h=w(k_1,k_2,\dots ,k_n)$

where:

$k_i=w_i(\text{elements of }G)$

then:

$h=(w\circ (w_1,w_2,\dots ,w_n))(\text{elements of }G)$