# Verbal subgroup of nilpotent group

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: nilpotent 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 subgroup of a group is termed a verbal subgroup of nilpotent group if the whole group is a nilpotent group and the subgroup is a verbal subgroup.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
verbal subgroup of abelian group |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant subgroup of nilpotent group the subgroup is a fully invariant subgroup and the whole group is nilpotent follows from verbal implies fully invariant follows from fully invariant not implies verbal in finite abelian group |FULL LIST, MORE INFO
characteristic subgroup of nilpotent group the subgroup is a characteristic subgroup and the whole group is nilpotent (via fully invariant) (via fully invariant) Fully invariant subgroup of nilpotent group|FULL LIST, MORE INFO
normal subgroup of nilpotent group the subgroup is a normal subgroup and the whole group is nilpotent (via characteristic) (via characteristic) Characteristic subgroup of nilpotent group, Fully invariant subgroup of nilpotent group|FULL LIST, MORE INFO
subgroup of nilpotent group the whole group is nilpotent Fully invariant subgroup of nilpotent group|FULL LIST, MORE INFO
verbal subgroup generated by a set of words any non-nilpotent group as a subgroup of itself |FULL LIST, MORE INFO
fully invariant subgroup invariant under all endomorphisms Fully invariant subgroup of nilpotent group|FULL LIST, MORE INFO
characteristic subgroup invariant under all automorphisms |FULL LIST, MORE INFO
normal subgroup invariant under all inner automorphisms |FULL LIST, MORE INFO