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 $H$.

Equivalence of definitions

• The equivalence of definitions (1) and (2) follows from the finite NPC theorem.
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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

See normal subgroup#Examples.

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
Finite normal subgroup				|FULL LIST, MORE INFO
Finitely generated normal subgroup
Normal closure of finite subset
Normal subgroup of finite index
Normal subgroup of periodic group
Normal subgroup of finitely generated group
Normal subgroup of slender group