Characteristic implies normal

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 must also satisfy the second subgroup property
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Statement

Verbal statement

Every characteristic subgroup of a group is a normal subgroup.

Symbolic statement

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

Property-theoretic statement

The subgroup property of being characteristic is stronger than the subgroup property of being normal.

Definitions used

Definitions for characteristic

A subgroup is characteristic if every automorphism of the whole group takes the subgroup to within itself. Thus, characteristicity is the invariance property with respect to the property of being an automorphism.
A subgroup is characteristic if every automorphism of the whole group restricts to an automorphism of the subgroup. Thus, characteristicity is expressible in the function restriction formalism as a property with both the left side and the right side being automorphism.
A subgroup is characteristic if any subgroup that resembles it via an automorphism is actually the same as it.
A subgroup is characteristic if it is a union of automorphism classes.

Definitions for normal

A subgroup is normal if every inner automorphism of the whole group takes the subgroup to within itself. Thus, normality is the invariance property with respect to the property of being an inner automorphism.
A subgroup is normal if every inner automorphism of the whole group restricts to an automorphism of the subgroup. Thus, normality is expressible in the function restriction formalism with the left side being inner automorphism and the right side being automorphism.
A subgroup is normal if any subgroup that resembles it as being a conjugate, is actually the same as it.
A subgroup is normal if it is a union of conjugacy classes.
A subgroup is normal if it is the kernel of a homomorphism of groups.

Proof

Hands-on proof

Let $H$ be a characteristic subgroup of $G$ . Then, for any automorphism $\sigma$ of $G$ , $\sigma (H)=H$ . This is in particularly true for the inner automorphisms. Thus, for any $g\in G$ , if $c_{g}$ denotes conjugation by $g$ , then $c_{g}(H)=H$ , or in other words, $gHg^{-1}=H$ . Thus $H$ is a normal subgroup of $G$ .

In terms of invariance properties

Normality is the invariance property with respect to inner automorphisms, and characteristicity is the invariance property with respect to automorphisms. Since the left side of normality implies the left side of characteristicity, every characteristic subgroup is normal.

In terms of resemblance

A subgroup is normal if any subgroup that resembles it as a conjugate is the same as it; a subgroup is characteristic if any subgroup that resembles it via an automorphism is the same as it. Since the notion of resembling as a conjugate is stronger than the notion of resembling via an automorphism, every characteristic subgroup is 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.

Converse

Normal subgroups need not be characteristic

Further information: Normal not implies characteristic

Normal-to-characteristic

A property that, along with normality, implies characterisicity, is termed a normal-to-characteristic subgroup property. A typical example is the property of being automorph-conjugate. This property plays on the resemblance-based definitions of normality and characteristicity.

Groups where every normal subgroup is characteristic

Further information: N=C-group A group in which the normal subgroups are precisely the same as the characteristic subgroups, is termed a N=C-group.

Intermediate properties

The proof that characteristic implies normal does not get simplified by splitting across intermediate properties. However, there are a number of properties lying between characteristicity and normality. Here are some of them:

Potentially characteristic subgroup is a subgroup that is characteristic in some group containing the bigger group.
Strongly potentially characteristic subgroup is a subgroup such that there is a group containing the bigger group in which both of them are characteristic.
Extensible automorphism-invariant subgroup is a subgroup that is invariant under every extensible automorphism (a property of automorphisms that is weaker than the property of being an inner automorphism)