# Commutator-verbal subgroup

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

## Definition

A subgroup of a group is a **commutator-verbal subgroup** if there is a set of words, each having the format of a commutator involving its letters and their inverses, such that the subgroup is precisely the subgroup generated by all the elements of the group expressible using those words.

## Examples

- All members of the derived series are commutator-verbal subgroups.
- All members of the lower central series are commutator-verbal subgroups.
- The subgroup generated by all elements of the form is a commutator-verbal subgroup.
- For a group , the subgroup is a commutator-verbal subgroup.

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Iterated commutator subgroup | obtained starting from the whole group through a series of operations whereby at each step we can take the commutator of two subgroups already there | iterated commutator subgroup implies commutator-verbal | commutator-verbal not implies iterated commutator subgroup | |FULL LIST, MORE INFO |

* Member of lower central series | ||||

* Member of derived series |

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Verbal subgroup | given by words | commutator-verbal implies verbal | verbal not implies commutator-verbal | |FULL LIST, MORE INFO |

## Metaproperties

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

### Commutator-closedness

This subgroup property is commutator-closed: the commutator of two subgroups each with the property, also has the property.

View other commutator-closed subgroup properties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity