# 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 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:

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 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) |