Verbality is quotient-transitive

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

Suppose $G$ is a group, with $H\le K\le G$ subgroups. Suppose $H$ is a verbal subgroup of $G$. Note that since verbal implies normal, $H$ is a normal subgroup of $G$, so that we can talk of the quotient group $G/H$. Suppose we are further given that $K/H$ is a verbal subgroup of $G/H$.

Then, $K$ must be a verbal subgroup of $G$. In fact, the set of words that we can use to generate $K$ is obtained by taking products of the set of words used for $H$ in $G$ and the set of words used for $K/H$ in $G/H$.

Related facts

• Verbality is transitive