Verbal subgroup

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This article defines a subgroup property related to (or which arises in the context of): geometric group theory
View other subgroup properties related to geometric group theory|View other terms related to geometric group theory | View all subgroup properties
This article defines a subgroup property related to (or which arises in the context of): combinatorial group theory
|View other terms related to combinatorial group theory | View all subgroup properties

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

A subgroup of 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 be a subvariety of the variety of groups. The verbal subgroup corresponding to is the unique smallest normal subgroup of such that . is a verbal subgroup of 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 .

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., (if using left action convention) abelian groups
member of lower central series the member corresponds to the word nilpotent groups of class at most
member of derived series commutator of two words, each of which is a commutator of two words, and so on, done times. Total of variables. solvable groups of derived length at most .
Subgroups generated by powers, for fixed , are also examples of verbal subgroups. all products of 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 powers for different integer values of . Note that gives the trivial subgroup and 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 nontrivial semidirect product of Z4 and Z4 Nontrivial semidirect product of Z4 and Z4 Klein four-group Klein four-group
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
Derived subgroup of nontrivial semidirect product of Z4 and Z4 Nontrivial semidirect product of Z4 and Z4 Cyclic group:Z2 Direct product of Z4 and Z2

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 central product of D8 and Z4 Central product of D8 and Z4 Cyclic group:Z4 Klein four-group
Center of direct product of D8 and Z2 Direct product of D8 and Z2 Klein four-group Klein four-group
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4 Nontrivial semidirect product of Z4 and Z4 Cyclic group:Z2 Quaternion 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
Q8 in central product of D8 and Z4 Central product of D8 and Z4 Quaternion group Cyclic group:Z2


Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property Yes verbality is transitive If are groups such that is a verbal subgroup of and is a verbal subgroup of , then is a verbal subgroup of .
quotient-transitive subgroup property Yes verbality is quotient-transitive If are groups such that is a verbal subgroup of and is a verbal subgroup of , then is a verbal subgroup of .
finite direct power-closed subgroup property Yes verbality is finite direct power-closed If is a verbal subgroup of and is a positive integer, then in the direct power , the subgroup is verbal.
direct power-closed subgroup property No verbality is not direct power-closed It is possible to have a group , a verbal subgroup of , and a cardinal such that the direct power is not verbal in .
finite-intersection-closed subgroup property No verbality is not finite-intersection-closed it is possible to have a group and verbal subgroups of such that the intersection is not verbal.
strongly join-closed subgroup property Yes verbality is strongly join-closed Suppose is a group and are all verbal subgroups of . Then, the join of subgroups is also a verbal subgroup of .
image condition Yes verbality satisfies image condition Suppose is a group and is a verbal subgroup. Suppose is a homomorphism of groups. Then, is a verbal subgroup of .

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 |
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 |
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 |
quasiverbal subgroup similar definition, replacing "variety" with "quasivariety" (see quasivarietal group property) |
pseudoverbal subgroup similar definition, replacing "variety" with "pseudovariety" (see pseudovarietal group property) |
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 |
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