# Hereditarily characteristic subgroup

From Groupprops

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