Characteristic implies normal

Statement
Let $$H$$ be a characteristic subgroup of $$G$$. Then, $$H$$ is normal in $$G$$.

Characteristic subgroup
The definitions we use here are as follows:


 * Hands-on definition: A subgroup $$H$$ of a group $$G$$ is termed a characteristic subgroup, if for any automorphism $$\sigma$$ of $$G$$, we have $$\sigma(H) = H$$.
 * Definition using function restriction expression: We can write characteristicity as the invariance property with respect to automorphisms:

Characteristic = Automorphism $$\to$$ 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.
 * Definition using relation implication expression: We can write characteristicity as:

Automorphic subgroups $$\implies$$ 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
The definitions we use here are as follows:


 * Hands-on definition: A subgroup $$H$$ of a group $$G$$ is termed normal, if for any $$g \in G$$, the inner automorphism $$c_g$$ defined by conjugation by $$g$$, namely the map $$x \mapsto gxg^{-1}$$, gives an isomorphism on $$H$$. In other words, for any $$g \in G$$:

$$c_g(H) = H$$

or more explicitly:

$$gHg^{-1} = H$$

Implicit in this definition is the fact that $$c_g$$ is an automorphism.


 * Definition using function restriction expression: We can write normality as the invariance property with respect to inner automorphisms:

Normal = Inner automorphism $$\to$$ Function

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


 * Definition using relation implication expression: We can write:

Normal = Conjugate subgroups $$\implies$$ 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.

Related facts about groups

 * Characteristic of normal implies normal: A characteristic subgroup of a normal subgroup is normal in the whole group.
 * Left transiter of normal is characteristic: In fact, characteristicity is precisely the property needed to be a left transiter for normality. Explicitly, if $$H \le K$$ is a subgroup such that whenever $$K$$ is normal in a group $$G$$, so is $$H$$, then $$H$$ mustbe characteristic in $$G$$.
 * Normal not implies characteristic: A normal subgroup of a group need not be characteristic in the group.

Analogues in other algebraic structures

 * Derivation-invariant implies ideal is the analogue for Lie rings.

Facts used

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

Hands-on proof
Given: $$H$$ is a characteristic subgroup of $$G$$. In other words, for any automorphism $$\sigma$$ of $$G$$, $$\sigma(H) = H$$.

To prove: For any $$g \in G$$, $$gHg^{-1} = H$$. In other words, if $$c_g$$ denotes conjugation by $$g$$ i.e. the map $$x \mapsto gxg^{-1}$$, then $$c_g(H) = H$$

Proof: $$c_g$$ is an inner automorphism, so it is an automorphism. Thus, invoking characteristicity, we have $$c_g(H) = H$$, i.e. $$gHg^{-1} =H$$.

Thus $$H$$ is a normal subgroup of $$G$$.

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

Normal = Inner automorphism $$\to$$ Function

Characteristic = Automorphism $$\to$$ Function

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

The relation implication expression for normality is:

Conjugate subgroups $$\implies$$ Equal

The relation implication expression for characteristicity is:

Automorphic subgroups $$\implies$$ 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.

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:


 * Monomial automorphism-invariant subgroup is a subgroup that is invariant under every monomial automorphism. It turns out that monomial automorphism-invariant subgroups are the same as normal subgroups.
 * Normal-extensible automorphism-invariant subgroup is a subgroup that is invariant under every normal-extensible automorphism.
 * Characteristic-extensible automorphism-invariant subgroup is a subgroup that is invariant under every characteristic-extensible automorphism.
 * IA-automorphism-invariant subgroup is a subgroup that is invariant under all IA-automorphisms.
 * Cofactorial automorphism-invariant subgroup is a subgroup that is invariant under all cofactorial automorphisms.

Properties involving additional structure
Invariance properties with respect to automorphisms that preserve additional structure imposed on the group, lie between characteristicity and normality. For instance:


 * Topologically characteristic subgroup in a topological group

Other intermediate properties

 * Potentially characteristic equals normal: A potentially characteristic subgroup is a subgroup that is characteristic in some bigger group containing the ambient group. A subgroup of a group is potentially characteristic iff it is normal.
 * Characteristic-potentially characteristic subgroup is a subgroup such that there is a group containing the bigger group in which both of them are characteristic.
 * Normal-potentially characteristic subgroup is a subgroup such that there is a group containing the bigger group as a normal subgroup and the smaller group as a characteristic subgroup.

Textbook references

 * , Page 17
 * , Page 135, Page 137 (Problem 6)
 * , Page 70, Problem 7(a)
 * , Page 28, 1.5.6(i)
 * , Page 234, Exercise 7 of Section 8 (Generators and relations)
 * , Page 4, Section 1.1 (statement and proof as passing mention in paragraph)