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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Get more facts about characteristic subgroup|Get more facts about normal subgroup

To learn more about the similarities and differences between characteristicity and normality, refer the survey article Characteristic versus normal

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.

Related facts

Characteristic of normal implies normal

Further information: Characteristic of normal implies normal

A characteristic subgroup of a normal subgroup, is normal in the whole group.

Left transiter of normal is characteristic

Further information: Left transiter of normal is characteristic

In fact, characteristicity is precisely the property needed to be a left transiter for normality. Explicitly, if $H \leq K$ is a subgroup such that whenever $K$ is normal in a group $G$ , so is $H$ , then $H$ mustbe characteristic in $G$ .

Definitions used

Characteristic subgroup

Further information: 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 $σ$ of $G$ , we have $σ (H) = H$ .
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 $⟹$ 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 $H$ 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 $x \mapsto g x g^{- 1}$ , gives an isomorphism on $H$ . In other words, for any $g \in G$ :

$c_{g} (H) = H$

or more explicitly:

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

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

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 $⟹$ 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

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

Proof

Hands-on proof

Given: $H$ is a characteristic subgroup of $G$ . In other words, for any automorphism $σ$ of $G$ , $σ (H) = H$ .

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

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

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 $\to$ Function

Characteristic = Automorphism $\to$ 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.

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.

A subgroup is termed an intermediately normal-to-characteristic subgroup if it is characteristic in every intermediate subgroup in which it is normal. Any intermediately automorph-conjugate subgroup, for instance, is intermediately normal-to-characteristic. All Sylow subgroups and Hall subgroups are intermediately normal-to-characteristic.

Groups where every normal subgroup is characteristic

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

Intermediate properties

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

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 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.

References

Textbook references

Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 17
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 135, Page 137 (Problem 6)
Topics in Algebra by I. N. Herstein, ^{More info}, Page 70, Problem 7(a)
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 28, 1.5.6(i)
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 234, Exercise 7 of Section 8 (Generators and relations)
Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, ^{More info}, Page 4, Section 1.1 (statement and proof as passing mention in paragraph)