Verbal subgroup of free group

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This article describes a property that arises as the conjunction of a subgroup property: verbal subgroup with a group property imposed on the ambient group: free group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
This article describes a property that arises as the conjunction of a subgroup property: fully invariant subgroup with a group property imposed on the ambient group: free group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

A verbal subgroup of free group is a subgroup of a group satsifying the following equivalent conditions:

  1. It is a verbal subgroup in the whole group and the whole group is a free group.
  2. It is a fully invariant subgroup in the whole group and the whole group is a free group.

Equivalence of definitions

For full proof, refer: fully invariant implies verbal in reduced free group

Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Characteristic subgroup of free group |FULL LIST, MORE INFO
Normal subgroup of free group |FULL LIST, MORE INFO
Subgroup of free group |FULL LIST, MORE INFO