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)
View all subgroup property implications | View all subgroup property non-implications
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

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

Definitions used

Characteristic subgroup

Further information: Characteristic subgroup

The definitions we use here are as follows:

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.

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

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 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. Further information: Group acts as automorphisms by conjugation

Normal = Inner automorphism \to Function

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

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

Related facts about groups

Analogues in other algebraic structures

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: 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.

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 \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.

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:

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:

Other intermediate properties

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)