# Powering-invariant central subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: powering-invariant subgroup and central subgroup
View other subgroup property conjunctions | view all subgroup properties

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

A subgroup $H$ of a group $G$ is termed a powering-invariant central subgroup if it satisfies the following equivalent conditions:

1. $H$ is both a powering-invariant subgroup of $G$ and a central subgroup of $G$.
2. $H$ is both a quotient-powering-invariant subgroup of $G$ and a central subgroup of $G$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
powering-invariant subgroup of abelian group

### Weaker properties

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