# 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.
- The whole group is a finite group and the subgroup is a purely definable subgroup of it, i.e,, the subgroup is definable in the first-order theory of the group.
- The whole group is a finite group and the subgroup is an elementarily characteristic subgroup of it, i.e., there is no other elementarily equivalently embedded subgroup.
- The whole group is a finite group and the subgroup is a monadic second-order characteristic subgroup of it.

## Relation with other properties

### Stronger properties

Property | Meaning | 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 | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal subgroup of finite group | ||||

finite characteristic subgroup | ||||

finite normal subgroup |