Verbal subgroup
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 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., ![]() |
abelian groups |
![]() |
the ![]() ![]() |
nilpotent groups of class at most ![]() |
![]() |
commutator of two words, each of which is a commutator of two words, and so on, done ![]() ![]() |
solvable groups of derived length at most ![]() |
Subgroups generated by ![]() ![]() |
all products of ![]() |
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:
Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:
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:
Here are some examples of subgroups in even more complicated/less basic groups not satisfying the property:
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
transitive subgroup property | Yes | verbality is transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
quotient-transitive subgroup property | Yes | verbality is quotient-transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
finite direct power-closed subgroup property | Yes | verbality is finite direct power-closed | If ![]() ![]() ![]() ![]() ![]() |
direct power-closed subgroup property | No | verbality is not direct power-closed | It is possible to have a group ![]() ![]() ![]() ![]() ![]() ![]() |
finite-intersection-closed subgroup property | No | verbality is not finite-intersection-closed | it is possible to have a group ![]() ![]() ![]() ![]() |
strongly join-closed subgroup property | Yes | verbality is strongly join-closed | Suppose ![]() ![]() ![]() ![]() ![]() |
image condition | Yes | verbality satisfies image condition | Suppose ![]() ![]() ![]() ![]() ![]() |
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) |