# Normal subgroup of finite group

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property imposed on the ambient 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

## Definition

A subgroup $H$ of a group $G$ is termed a normal subgroup of finite group if it satisfies the following equivalent conditions:

1. The whole group $G$ is a finite group and the subgroup $H$ is a normal subgroup of it.
2. There exists a finite group $K$ containing $G$ such that $H$ is a characteristic subgroup of $K$.
3. $G$ is a finite group and there is a generating set $A$ for $H$ and a generating set $B$ for $G$ such that the conjugate of any element of $A$ by any element of $B$ is in $H$.

## Examples

VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups whose group part dissatisfies property finite group
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## Relation with other properties

### How it differs from normality in general

The property of normality inside a finite group is largely similar to the property of normality in an arbitrary group -- most of the property implications, metaproperties, etc. hold for finite groups. There are some small differences:

• To check that a subgroup of a finite group is normal, it suffices to check that it is sent to within itself under all elements in a generating set of the whole group. For an infinite group, we need to check that it is sent to within itself by all elements of a generating set and their inverses.
• Normal subgroups of finite groups, and more generally finite normal subgroups, behave nicely with respect to amalgams: finite normal implies amalgam-characteristic. This is not true for normal subgroups in arbitrary groups.

### Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Characteristic subgroup of finite group characteristic implies normal normal not implies characteristic |FULL LIST, MORE INFO
Fully invariant subgroup of finite group
Isomorph-free subgroup of finite group
Subgroup of finite abelian group
Normal subgroup of group of prime power order

### Weaker properties

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