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)	|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)	|FULL LIST, MORE INFO
subgroup of nilpotent group	the whole group is nilpotent			|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			|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