# Finite normal subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): finite group

View a complete list of such conjunctions

## Contents

## Definition

A subgroup of a group is termed a **finite normal subgroup** if it is finite (as a group) and normal as a subgroup.

## Examples

VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups dissatisfying property finite groupVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## Relation with other properties

### Stronger properties

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

normal subgroup of prime order | normal subgroup and its order is a prime number | |||

finite central subgroup | finite and a central subgroup | central implies normal | any of the finite examples for normal not implies central | |FULL LIST, MORE INFO |

finite characteristic subgroup | finite and a characteristic subgroup | any of the finite examples for normal not implies characteristic | |FULL LIST, MORE INFO | |

normal subgroup of finite group | normal and the whole group is finite. | |FULL LIST, MORE INFO |