# Hereditarily characteristic subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This article describes a property that arises as the conjunction of a subgroup property: normal-hereditarily characteristic subgroup with a group property (itself viewed as a subgroup property): Dedekind group
View a complete list of such conjunctions

## Definition

A subgroup of a group is termed a hereditarily characteristic subgroup if every subgroup of it is a characteristic subgroup in the whole group.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Cyclic characteristic subgroup cyclic group and characteristic subgroup of whole group cyclic characteristic implies hereditarily characteristic hereditarily characteristic not implies cyclic in finite

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal-hereditarily characteristic subgroup every normal subgroup of the subgroup is characteristic in the whole group
Hereditarily normal subgroup every subgroup is normal in the whole group

## Metaproperties

### Left realization

For a group $G$ to be realizable as a hereditarily characteristic subgroup in a bigger group $K$, a necessary condition is that $G$ be a Dedekind group, i.e., every subgroup of $G$ must be normal in $G$.