Verbal subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

How verbal subgroups came about

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

Definition

Symbol-free definition

A verbal subgroup is a subgroup generated by the set of elements spanned by the values taken by a collection of words with parameters, as the parameters vary over all elements of the group.

Definition with symbols

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 $G$ . A subgroup $H$ of $G$ is termed verbal if it is generated by the span of a collection of words.

Examples

Typical examples of verbal subgroups

The commutator subgroup, as well as all members of the derived series as well as of the lower central series, are verbal subgroups. In fact, these are very special kinds of verbal subgroups: each of them can be obtained as the span of only a single word.

For the commutator subgroup, this word is the commutator of two unknowns, that is, it is the word $x y x^{- 1} y^{- 1} = [x, y]$ .

Subgroups generated by $n^{t h}$ powers, for fixed $n$ , are also examples of verbal subgroups.

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

Relation with other properties

Stronger properties

Commutator-verbal subgroup is a verbal subgroup where all the words are described in terms of the commutator symbol.
Member of the lower central series (finite part)
Member of the derived series (finite part)

Weaker properties

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Verbality is transitive. The proof of this rests on the observation that we can actually expand the collection $C$ of words to a new collection $C^{'}$ such that the elements of the subgroup are precisely those realized by words in $C^{'}$ .

Verbality is also identity-true. It is in fact a trim property.

Proof that verbality is transitive:

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

Is an intersection of verbal subgroups verbal?

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

The subgroup generated by a family of verbal subgroups is indeed verbal. In fact, the collection of words for the subgroup generated is simply the union of the collection of words for each of the subgroups.