# Powering-invariant characteristic subgroup

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

## Definition

A subgroup of a group is termed a powering-invariant characteristic subgroup if it is both a powering-invariant subgroup and a characteristic subgroup.

Note that characteristic not implies powering-invariant. Also, a non-characteristic subgroup of a finite group is powering-invariant and not characteristic. Hence, neither property implies the other, so the conjunction is strictly stronger than both.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
quotient-powering-invariant characteristic subgroup
characteristic subgroup of abelian group characteristic subgroup of abelian group is powering-invariant Characteristic subgroup of center, Quotient-powering-invariant characteristic subgroup|FULL LIST, MORE INFO
complemented characteristic subgroup (via quotient-powering-invariant characteristic) Quotient-powering-invariant characteristic subgroup|FULL LIST, MORE INFO
finite characteristic subgroup (via quotient-powering-invariant characteristic) Quotient-powering-invariant characteristic subgroup|FULL LIST, MORE INFO
characteristic subgroup of finite index (via quotient-powering-invariant characteristic) Quotient-powering-invariant characteristic subgroup|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
powering-invariant normal subgroup [powering-invariant subgroup]] as well as a normal subgroup follows from characteristic implies normal follows from finite examples for normal not implies characteristic |FULL LIST, MORE INFO