# Normal subgroup

In group theory, a branch of mathematics, a **normal subgroup**, also known as **invariant subgroup**, or **normal divisor**, is a (proper or improper) subgroup *H* of the group *G* that is invariant under conjugation by all elements of *G*.

Two elements, *a′* and *a*, of *G* are said to be conjugate by *g* ∈ *G*, if
*a′* = *g a g ^{−1}*. Clearly,

*a*=

*g*, so that conjugation is symmetric;

^{−1}a′ g*a*and

*a′*are conjugate partners.

If for all *h* ∈ *H* and all *g* ∈ *G* it holds that: *g h g ^{−1}* ∈

*H*, then

*H*is a normal subgroup of

*G*, (also expressed as "

*H*is invariant in

*G*"). That is, with

*h*in

*H*all conjugate partners of

*h*are also in

*H*.

## Equivalent definitions

A subgroup *H* of a group *G* is termed **normal** if the following equivalent conditions are satisfied:

- Given any and , we have
*H*occurs as the kernel of a homomorphism from*G*. In other words, there is a homomorphism such that the inverse image of the identity element of*K*is*H*.- Every inner automorphism of
*G*sends*H*to within itself - Every inner automorphism of
*G*restricts to an automorphism of*H* - The left cosets and right cosets of
*H*are always equal: . (This is often expressed as: "*H*is simultaneously left- and right-invariant").

## Some elementary examples and counterexamples

### Klein's Vierergruppe in *S*_{4}

The set of all permutations of 4 elements forms the symmetric group *S*_{4}, which is of order of 4! = 24. The group of the following four permutations is a subgroup and has the structure of Felix Klein's Vierergruppe:

*V*_{4}≡ {(1), (12)(34), (13)(24), (14)(23)}

It is easily verified that *V*_{4} is a normal subgroup of *S*_{4}. [Conjugation preserves the cycle structure (..)(..) and *V*_{4} contains all elements with this structure.]

### All subgroups in Abelian groups

In an Abelian group, every subgroup is normal. This is because if is an Abelian group, and , then .

More generally, any subgroup inside the center of a group is normal.

It is not, however, true that if every subgroup of a group is normal, then the group must be Abelian. A counterexample is the quaternion group.

### All characteristic subgroups

A characteristic subgroup of a group is a subgroup which is invariant under all automorphisms of the whole group. Characteristic subgroups are normal, because normality requires invariance only under inner automorphisms, which are a particular kind of automorphism.

In particular, subgroups like the center, the commutator subgroup, the Frattini subgroup are examples of characteristic, and hence normal, subgroups.

### A smallest counterexample

The smallest example of a non-normal subgroup is a subgroup of order two in the symmetric group on three elements. Explicitly, we can take the cyclic subgroup of order two generated by the 2-cycle in the symmetric group of permutations on symbols .

## Properties

The intersection of any family of normal subgroups is again a normal subgroup. We can therefore define the normal subgroup *generated* by a subset *S* of a group *G* to be the intersection of all normal subgroups of *G* containing *S*.

## Quotient group

The **quotient group** of a group *G* by a normal subgroup *N* is defined as the set of (left or right) cosets:

with the the group operations

and the coset as identity element. It is easy to check that these define a group structure on the set of cosets and that the **quotient map** is a group homomorphism. Because of this property *N* is sometimes called a *normal divisor* of *G*.

### First Isomorphism Theorem

The First Isomorphism Theorem for groups states that if is a group homomorphism then the kernel of *f*, say *K*, is a normal subgroup of *G*, and the map *f* factors through the quotient map and an injective homomorphism *i*:

## External links

- Normal subgroup on Mathworld
- Normal subgroup on Planetmath