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 subgroup properties related to 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 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 partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2

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

 Group partSubgroup partQuotient part
A4 in S4Symmetric group:S4Alternating group:A4Cyclic group:Z2
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Center of special linear group:SL(2,3)Special linear group:SL(2,3)Cyclic group:Z2Alternating group:A4
Center of special linear group:SL(2,5)Special linear group:SL(2,5)Cyclic group:Z2Alternating group:A5
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Klein four-group
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric group:S3

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

 Group partSubgroup partQuotient part
Center of M16M16Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-groupKlein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Derived subgroup of M16M16Cyclic group:Z2Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z4Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct 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 partSubgroup partQuotient part
S2 in S3Symmetric group:S3Cyclic group:Z2
Z2 in V4Klein four-groupCyclic group:Z2Cyclic group:Z2

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

 Group partSubgroup partQuotient part
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4Cyclic group:Z2
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-groupCyclic group:Z2
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

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

 Group partSubgroup partQuotient part
Center of central product of D8 and Z4Central product of D8 and Z4Cyclic group:Z4Klein four-group
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Quaternion group
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8Cyclic group:Z2
D8 in D16Dihedral group:D16Dihedral group:D8Cyclic group:Z2
D8 in SD16Semidihedral group:SD16Dihedral group:D8Cyclic group:Z2
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2Cyclic group:Z2
Klein four-subgroup of M16M16Klein four-groupCyclic group:Z4
Non-central Z4 in M16M16Cyclic group:Z4Cyclic group:Z4
Q8 in SD16Semidihedral group:SD16Quaternion groupCyclic group:Z2
Q8 in central product of D8 and Z4Central product of D8 and Z4Quaternion groupCyclic 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