Fully invariant subgroup of nilpotent group

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: 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 fully invariant subgroup of nilpotent group if the whole group is a nilpotent group and the subgroup is a fully invariant subgroup.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant subgroup of abelian group abelian implies nilpotent follows from nilpotent not implies abelian (use the group as a subgroup of itself) |FULL LIST, MORE INFO
verbal subgroup of nilpotent group follows from verbal implies fully invariant follows from fully invariant not implies verbal in finite abelian group |FULL LIST, MORE INFO