# Characteristic implies normal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) must also satisfy the second subgroup property (i.e., normal subgroup)
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To learn more about the similarities and differences between characteristicity and normality, refer the survey article Characteristic versus normal

## Statement

Let  be a characteristic subgroup of . Then,  is normal in .

## Definitions used

### Characteristic subgroup

Further information: Characteristic subgroup

The definitions we use here are as follows:

Characteristic = Automorphism  Function

This is interpreted as: any automorphism from the whole group to itself, restricts to a function from the subgroup to itself. In other words, the subgroup is invariant under automorphisms.

Automorphic subgroups  Equal

In other words, any subgroup obtained by taking the image of this subgroup under an automorphism of the whole group, must be equal to it.

• Definition in terms of equivalence classes of elements: A subgroup is characteristic if and only if it is the union of equivalence classes of elements under the action of the automorphism group.

### Normal subgroup

Further information: Normal subgroup

The definitions we use here are as follows:

• Hands-on definition: A subgroup  of a group  is termed normal, if for any , the inner automorphism  defined by conjugation by , namely the map , gives an isomorphism on . In other words, for any :



or more explicitly:



Implicit in this definition is the fact that  is an automorphism. Further information: Group acts as automorphisms by conjugation

Normal = Inner automorphism  Function

In other words, any inner automorphism on the whole group restricts to a function from the subgroup to itself.

Normal = Conjugate subgroups  Equal

In other words, any subgroup conjugate to the given one, must be equal to it.

• Definition using equivalence classes of elements: A subgroup is normal if and only if it is a union of conjugacy classes of elements.

## Facts used

1. Group acts as automorphisms by conjugation: This states that every inner automorphism of a group is an automorphism.

## Proof

### Hands-on proof

Given:  is a characteristic subgroup of . In other words, for any automorphism  of , .

To prove: For any , . In other words, if  denotes conjugation by  i.e. the map , then 

Proof:  is an inner automorphism, so it is an automorphism. Thus, invoking characteristicity, we have , i.e. .

Thus  is a normal subgroup of .

### Using function restriction expressions

This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic

Normality is the invariance property with respect to inner automorphisms, and characteristicity is the invariance property with respect to automorphisms. Explicitly:

Normal = Inner automorphism  Function

Characteristic = Automorphism  Function

Since the left side of normality implies the left side of characteristicity, every characteristic subgroup is normal.

### Using relation implication expressions

This subgroup property implication can be proved using relation implication expressions for the subgroup properties
View other implications proved in this way OR Read a survey article on the topic

The relation implication expression for normality is:

Conjugate subgroups  Equal

The relation implication expression for characteristicity is:

Automorphic subgroups  Equal

Since the left side for the expression for normality is stronger than the left side for the expression for characteristicity, and the right sides are the same, the subgroup property of being characteristic implies the subgroup property of being normal.

### In terms of equivalence classes of elements

A normal subgroup is a subgroup that is a union of conjugacy classes; a characteristic subgroup is a subgroup that is a union of automorphism classes. Since every automorphism class is a union of conjugacy classes, every characteristic subgroup is normal.

## Related properties

### Normal-to-characteristic

Further information: Subnormal-to-normal and normal-to-characteristic

A normal-to-characteristic subgroup is a subgroup that, if normal, is also characteristic. An intermediately normal-to-characteristic subgroup is a subgroup that, if normal in any intermediate subgroup, is also characteristic in that intermediate subgroup. There are a number of subgroup properties that are stronger than the property of being normal-to-characteristic: in other words, any normal subgroup satisfying the property is also characteristic.

These include, for instance, the property of being automorph-conjugate, procharacteristic, paracharacteristic, core-characteristic, closure-characteristic, and many others. For more information on such properties, refer subnormal-to-normal and normal-to-characteristic.

### Intermediate invariance properties

The invariance property for an automorphism property that is weaker than the property of being an inner automorphism, lies somewhere between characteristicity and normality. Here are some examples: