Verbal subgroup

This article defines a subgroup property related to (or which arises in the context of): geometric group theory
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This article defines a subgroup property related to (or which arises in the context of): combinatorial group theory
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History

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

Definition

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.

Equivalence of definitions

Further information: equivalence of definitions of verbal subgroup

Examples

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

Verbal subgroup	Corresponding word or words	Corresponding variety of groups
derived subgroup (also called abelianization)	commutator of two elements, i.e., $[x,y] = xyx^{-1}y^{-1}$ (if using left action convention)	abelian groups
$(c+1)^{th}$ member of lower central series	the $(c+1)^{th}$ member corresponds to the word $[[\dots[x_1,x_2],\dots,x_{c+1}]$	nilpotent groups of class at most $c$
$\ell^{th}$ member of derived series	commutator of two words, each of which is a commutator of two words, and so on, done $\ell$ times. Total of $2^\ell$ variables.	solvable groups of derived length at most $\ell$.
Subgroups generated by $n^{th}$ powers, for fixed $n$, are also examples of verbal subgroups.	all products of $n^{th}$ powers

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. For full proof, refer: Verbal subgroup equals power subgroup in abelian group

Examples of subgroups satisfying the property

Here are some examples of subgroups in basic/important groups satisfying the property:

	Group part	Subgroup part	Quotient part
A3 in S3	Symmetric group:S3	Cyclic group:Z3	Cyclic group:Z2

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

	Group part	Subgroup part	Quotient part
A4 in S4	Symmetric group:S4	Alternating group:A4	Cyclic group:Z2
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of quaternion group	Quaternion group	Cyclic group:Z2	Klein four-group
Center of special linear group:SL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2	Alternating group:A4
Center of special linear group:SL(2,5)	Special linear group:SL(2,5)	Cyclic group:Z2	Alternating group:A5
First agemo subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Klein four-group
Klein four-subgroup of alternating group:A4	Alternating group:A4	Klein four-group	Cyclic group:Z3
Normal Klein four-subgroup of symmetric group:S4	Symmetric group:S4	Klein four-group	Symmetric group:S3

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

	Group part	Subgroup part	Quotient part
Center of M16	M16	Cyclic group:Z4	Klein four-group
Center of dihedral group:D16	Dihedral group:D16	Cyclic group:Z2	Dihedral group:D8
Center of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z2	Dihedral group:D8
Derived subgroup of M16	M16	Cyclic group:Z2	Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z4	Klein four-group

Examples of subgroups not satisfying the property

Here are some examples of subgroups in basic/important groups not satisfying the property:

	Group part	Subgroup part	Quotient part
S2 in S3	Symmetric group:S3	Cyclic group:Z2
Z2 in V4	Klein four-group	Cyclic group:Z2	Cyclic group:Z2

Here are some some examples of subgroups in relatively less basic/important groups not satisfying the property:

	Group part	Subgroup part	Quotient part
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2
Cyclic maximal subgroups of quaternion group	Quaternion group	Cyclic group:Z4	Cyclic group:Z2
First omega subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Klein four-group	Cyclic group:Z2
Klein four-subgroups of dihedral group:D8	Dihedral group:D8	Klein four-group	Cyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Cyclic group:Z4
Z4 in direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z4	Cyclic group:Z2

Here are some examples of subgroups in even more complicated/less basic groups not satisfying the property:

	Group part	Subgroup part	Quotient part
Center of direct product of D8 and Z2	Direct product of D8 and Z2	Klein four-group	Klein four-group
Cyclic maximal subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z8	Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z8	Cyclic group:Z2
D8 in D16	Dihedral group:D16	Dihedral group:D8	Cyclic group:Z2
D8 in SD16	Semidihedral group:SD16	Dihedral group:D8	Cyclic group:Z2
Direct product of Z4 and Z2 in M16	M16	Direct product of Z4 and Z2	Cyclic group:Z2
Klein four-subgroup of M16	M16	Klein four-group	Cyclic group:Z4
Non-central Z4 in M16	M16	Cyclic group:Z4	Cyclic group:Z4
Q8 in SD16	Semidihedral group:SD16	Quaternion group	Cyclic group:Z2

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	verbality is transitive	If $H \le K \le G$ are groups such that $H$ is a verbal subgroup of $K$ and $K$ is a verbal subgroup of $G$, then $H$ is a verbal subgroup of $G$.
quotient-transitive subgroup property	Yes	verbality is quotient-transitive	If $H \le K \le G$ are groups such that $H$ is a verbal subgroup of $G$ and $K/H$ is a verbal subgroup of $G/H$, then $K$ is a verbal subgroup of $G$.
finite direct power-closed subgroup property	Yes	verbality is finite direct power-closed	If $H$ is a verbal subgroup of $G$ and $n$ is a positive integer, then in the direct power $G^n$, the subgroup $H^n$ is verbal.
direct power-closed subgroup property	No	verbality is not direct power-closed	It is possible to have a group $G$, a verbal subgroup $H$ of $G$, and a cardinal $\alpha$ such that the direct power $H^\alpha$ is not verbal in $G^\alpha$.
finite-intersection-closed subgroup property	No	verbality is not finite-intersection-closed	it is possible to have a group $G$ and verbal subgroups $H,K$ of $G$ such that the intersection $H \cap K$ is not verbal.
strongly join-closed subgroup property	Yes	verbality is strongly join-closed	Suppose $G$ is a group and $H_i, i \in I$ are all verbal subgroups of $G$. Then, the join of subgroups $\langle H_i \rangle$ is also a verbal subgroup of $G$.
image condition	Yes	verbality satisfies image condition	Suppose $G$ is a group and $H$ is a verbal subgroup. Suppose $\varphi:G \to K$ is a homomorphism of groups. Then, $\varphi(H)$ is a verbal subgroup of $K$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
commutator-verbal subgroup	a verbal subgroup where all the words are described in terms of the commutator symbol			|FULL LIST, MORE INFO
verbal subgroup of finite group	the whole group is a finite group			Verbal subgroup of finite type|FULL LIST, MORE INFO
verbal subgroup of finite type	it is a union of the images of finitely many word maps			Template:Inermediate notions short
verbal subgroup of finitely generated type	it is generated by the union of the images of finitely many word maps			|FULL LIST, MORE INFO
member of the lower central series (finite part)
member of the derived series (finite part)

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
intersection of finitely many verbal subgroups	can be expressed as an intersection of a finite number of verbal subgroups			|FULL LIST, MORE INFO
quasiverbal subgroup	similar definition, replacing "variety" with "quasivariety" (see quasivarietal group property)			|FULL LIST, MORE INFO
pseudoverbal subgroup	similar definition, replacing "variety" with "pseudovariety" (see pseudovarietal group property)			|FULL LIST, MORE INFO
existentially bound-word subgroup	subgroup described by a set of equations quantified existentially in all other variables	verbal implies existentially bound-word	existentially bound-word not implies verbal
image-closed fully invariant subgroup	under any surjective homomorphism from the whole group, the image of the subgroup is fully invariant in the target group	verbal implies image-closed fully invariant	image-closed fully invariant not implies verbal	|FULL LIST, MORE INFO
fully invariant subgroup (also called fully characteristic subgroup)	invariant under all endomorphisms	verbal implies fully invariant	fully invariant not implies verbal (see also list of examples)	Existentially bound-word subgroup, Image-closed fully invariant subgroup, Intersection of finitely many verbal subgroups, Pseudoverbal subgroup, Quasiverbal subgroup, Quotient-subisomorph-containing subgroup, Weakly image-closed fully invariant subgroup|FULL LIST, MORE INFO
image-closed characteristic subgroup	under any surjective homomorphism from the whole group, the image of the subgroup is characteristic in the target group	(via image-closed fully invariant)	(via image-closed fully invariant)	Image-closed fully invariant subgroup|FULL LIST, MORE INFO
characteristic subgroup	invariant under all automorphisms	(via fully invariant)	(via fully invariant)	Bound-word subgroup, Finite direct power-closed characteristic subgroup, Fully invariant subgroup, Image-closed characteristic subgroup, Image-closed fully invariant subgroup, Intersection of finitely many verbal subgroups, Pseudoverbal subgroup, Quasiverbal subgroup, Quotient-subisomorph-containing subgroup, Weakly image-closed characteristic subgroup, Weakly image-closed fully invariant subgroup|FULL LIST, MORE INFO
strictly characteristic subgroup	invariant under all surjective endomorphisms	(via fully invariant)	(via fully invariant)	Bound-word subgroup, Existentially bound-word subgroup, Fully invariant subgroup, Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms	(via characteristic)	(via characteristic)	Bound-word subgroup, Characteristic subgroup, Finite direct power-closed characteristic subgroup, Fully invariant subgroup, Image-closed characteristic subgroup, Intersection of finitely many verbal subgroups, Normal-potentially verbal subgroup, Potentially fully invariant subgroup, Weakly image-closed characteristic subgroup, Weakly image-closed fully invariant subgroup|FULL LIST, MORE INFO