Verbal subgroup

History
The notion of verbal subgroup was introduced in the study of free groups in combinatorial group theory.

Definition in terms of words and word maps
Let $$C$$ be a collection of words (or expressions in terms of the group operations, in unknown variables). Define the span of $$C$$ in a group $$G$$ as the collection of elements of $$G$$ which are realized from words in $$C$$ by substituting, for the variables, elements of $$G$$. In other words, the span of $$C$$ is defined as the union of the images of the word maps for every word in $$G$$.

A subgroup $$H$$ of $$G$$ is termed verbal if it satisfies the following equivalent conditions:


 * It is generated by the span of a collection of words
 * It is itself the span of a collection of words

Definition in terms of varieties
Let $$\mathcal{V}$$ be a subvariety of the variety of groups. The verbal subgroup corresponding to $$\mathcal{V}$$ is the unique smallest normal subgroup $$H$$ of $$G$$ such that $$G/H \in \mathcal{V}$$. $$H$$ is a verbal subgroup of $$\mathcal{V}$$ if it is a verbal subgroup corresponding to some subvariety of the variety of groups.

Extreme examples

 * The trivial subgroup is a verbal subgroup corresponding to the word that just gives the identity element.
 * The whole group is a verbal subgroup corresponding to the word in one letter that's just that letter, i.e., the word $$x$$.

Typical examples of verbal subgroups
Since every word is essentially a combination of commutator and power operations, these are somewhat representative examples of verbal subgroups.

In an abelian group
In an abelian group, the only verbal subgroups are the sets of $$n^{th}$$ powers for different integer values of $$n$$. Note that $$n = 0$$ gives the trivial subgroup and $$n = 1$$ gives the whole group.