Characteristic of normal implies normal

From Groupprops
Jump to: navigation, search
This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Characteristic subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., Normal subgroup (?))
View a complete list of composition computations

Statement

Property-theoretic statement

Characteristic * Normal Normal

Here, denotes the composition operator.

Verbal statement

Every Characteristic subgroup (?) of a Normal subgroup (?) is normal.

Statement with symbols

Let such that is characteristic in and is normal in , then is normal in .

Related facts

Basic ideas implicit in the definitions

  • Restriction of automorphism to subgroup invariant under it and its inverse is automorphism: If is a subgroup and is an automorphism of such that both and send to within itself, then restricts to an automorphism of . This is the key idea used in arguing that an inner automorphism of the biggest group must restrict to an automorphism of the intermediate subgroup, rather than merely to a homomorphism from the intermediate subgroup to itself. Note that this idea is implicit in the equivalence between different formulations of the notion of normal subgroup.

Related facts in group theory

Analogues

Applications

For a complete list of applications, refer:

Category:Applications of characteristic of normal implies normal

Definitions used

Characteristic subgroup

Further information: Characteristic subgroup

The definitions we use here are as follows:

Characteristic = Automorphism Automorphism

This is interpreted as: any automorphism from the whole group to itself, restricts to an automorphism from the subgroup to itself. Note that this is stronger than simply saying that it maps the subgroup to within itself -- we also demand that the restriction be an automorphism of the subgroup.

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 a map from to itself. 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

Note that it turns out that the above also implies that (This is because we have as well as ). This equivalence of ideas is crucial to the proof.

Normal = Inner automorphism Automorphism

In other words, any inner automorphism on the whole group restricts to an automorphism from the subgroup to itself. Note that this is stronger than saying that the inner automorphism simply sends the subgroup to itself -- we also demand that the restriction itself be an automorphism of the subgroup.

Facts used

  1. Restriction of automorphism to subgroup invariant under it and its inverse is automorphism
  2. Composition rule for function restriction: This is used for the proof using function restriction expressions.

Proof

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

Hands-on proof

Given: Groups such that is characteristic in and is normal in . An element .

To Prove: The map , the map maps to (and in fact, yields an automorphism of ).

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 for every and restricts to an automorphism of . Call this automorphism . definition of normal subgroup
Fact (1)
is normal in
is in .
[SHOW MORE]
2 sends to itself, and in fact restricts to an automorphism of . definition of characteristic subgroup is characteristic in Step (1) direct
3 sends to itself and restricts to an automorphism of . Steps (1), (2) [SHOW MORE]

Using function restriction expressions

In terms of the function restriction formalism:

Inner automorphism Automorphism

In other words, every inner automorphism of the whole group restricts to an automorphism of the subgroup.

Automorphism Automorphism

In other words, every automorphism of the whole group restricts to an automorphism of the subgroup.

We now use the composition rule for function restriction to observe that the composition of characteristic and normal implies the property:

Inner automorphism Automorphism

Which is again the subgroup property of normality.

References

Textbook references

External links

Search links