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 $H\leq K\leq G$ 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:

$k_{i}=w_{i}({\mbox{elements of }}G)$

then:

$h=(w\circ (w_{1},w_{2},\dots ,w_{n}))({\mbox{elements of }}G)$