# Quotient-powering-invariant characteristic subgroup

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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

## 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