Commutator-verbal subgroup

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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]

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 [[x,y],y] is a commutator-verbal subgroup.
  • For a group G, the subgroup [[G,G],[[G,G],G]] 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 transitive
ABOUT 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