# Quotient-powering-invariant characteristic subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: quotient-powering-invariant subgroup and characteristic subgroup

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a group is termed a **quotient-powering-invariant characteristic subgroup** if it is both a quotient-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 |
---|---|---|---|---|

characteristic subgroup of abelian group | characteristic subgroup of abelian group is quotient-powering-invariant | Characteristic subgroup of center|FULL LIST, MORE INFO | ||

complemented characteristic subgroup | follows from complemented normal implies quotient-powering-invariant | |FULL LIST, MORE INFO | ||

finite characteristic subgroup | finite normal implies quotient-powering-invariant and characteristic implies normal | |FULL LIST, MORE INFO | ||

characteristic subgroup of finite index | normal of finite index implies quotient-powering-invariant and characteristic implies normal | |FULL LIST, MORE INFO |

### Weaker properties

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

powering-invariant characteristic subgroup | |FULL LIST, MORE INFO | |||

quotient-powering-invariant subgroup | |FULL LIST, MORE INFO | |||

powering-invariant normal subgroup | Powering-invariant characteristic subgroup|FULL LIST, MORE INFO | |||

powering-invariant subgroup | Powering-invariant characteristic subgroup|FULL LIST, MORE INFO | |||

characteristic subgroup | Powering-invariant characteristic subgroup|FULL LIST, MORE INFO | |||

normal subgroup | Powering-invariant characteristic subgroup|FULL LIST, MORE INFO |