# Characteristic subgroup of finite group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on theambient group: finite group

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

A subgroup of a group is termed a **characteristic subgroup of finite group** if it satisfies the following equivalent conditions:

- The whole group is a finite group and the subgroup is a characteristic subgroup of it.
- The whole group is a finite group and the subgroup is a strictly characteristic subgroup(i.e., invariant under all surjective endomorphisms) of it.
- The whole group is a finite group and the subgroup is an injective endomorphism-invariant subgroup of it.

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Isomorph-free subgroup of finite group | ||||

Fully invariant subgroup of finite group | ||||

Normal Sylow subgroup | ||||

Normal Hall subgroup | ||||

Characteristic subgroup of group of prime power order |

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Normal subgroup of finite group | ||||

Finite characteristic subgroup | ||||

Finite normal subgroup |