# Local powering-invariant normal subgroup of nilpotent group

This article describes a property that arises as the conjunction of a subgroup property: local powering-invariant normal 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 termed a local powering-invariant normal subgroup of nilpotent group if it satisfies the following equivalent conditions:

1. $G$ is a nilpotent group and $H$ is a local powering-invariant normal subgroup of $G$, i.e., $H$ is both a local powering-invariant subgroup and a normal subgroup of $G$.
2. $G$ is a nilpotent group and $H$ is a quotient-local powering-invariant subgroup of $G$.
3. $G$ is a nilpotent group and $H$ is a quotient-torsion-freeness-closed subgroup of $G$.

## Relation with other properties

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions