Powering-invariant characteristic subgroup of nilpotent group

This article describes a property that arises as the conjunction of a subgroup property: powering-invariant characteristic 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 $H$ of a group $G$ is called a powering-invariant characteristic subgroup of nilpotent group if $G$ is a nilpotent group and $H$ is a powering-invariant characteristic subgroup, i.e., it is both a powering-invariant subgroup and a characteristic subgroup.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
divisibility-closed characteristic subgroup of nilpotent group the group is nilpotent and the subgroup is a divisibility-closed characteristic subgroup: it is both divisibility-closed and characteristic follows from divisibility-closed implies powering-invariant |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup of nilpotent group the group is nilpotent and the subgroup is a characteristic subgroup characteristic not implies powering-invariant in nilpotent group |FULL LIST, MORE INFO
powering-invariant normal subgroup of nilpotent group the group is nilpotent and the subgroup is a powering-invariant normal subgroup follows from characteristic implies normal any proper nonzero vector subspace of a nonzero vector space over the rationals |FULL LIST, MORE INFO