Characteristic of normal implies normal

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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 (?))
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Statement

Property-theoretic statement

Characteristic * Normal \le Normal

Here, * denotes the composition operator.

Verbal statement

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

Statement with symbols

Let H \le K \le G such that H is characteristic in K and K is normal in G, then H is normal in G.

Related facts

Basic ideas implicit in the definitions

  • Restriction of automorphism to subgroup invariant under it and its inverse is automorphism: If K \le G is a subgroup and \sigma is an automorphism of G such that both \sigma and \sigma^{-1} send K to within itself, then \sigma restricts to an automorphism of K. 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 \to 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 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 a map from H to itself. In other words, for any g \in G:

c_g(H) \le H

or more explicitly:

gHg^{-1} \le H

Implicit in this definition is the fact that c_g is an automorphism. Further information: Group acts as automorphisms by conjugation

Note that it turns out that the above also implies that c_g(H) = H (This is because we have c_g(H) \le H as well as c_{g^{-1}}(H) \le H). This equivalence of ideas is crucial to the proof.

Normal = Inner automorphism \to 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 H \le K \le G such that H is characteristic in K and K is normal in G. An element g \in G.

To Prove: The map g \in G, the map c_g : x \mapsto gxg^{-1} maps H to H (and in fact, yields an automorphism of H).

Proof:

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

Using function restriction expressions

In terms of the function restriction formalism:

Inner automorphism \to Automorphism

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

Automorphism \to 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 \to Automorphism

Which is again the subgroup property of normality.

References

Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 135, Section 4.4 (Automorphisms), Point (3) after definition of characteristic subgroup, More info. Also, Page 137, Exercise 8(a).
  • A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, Page 28, Section 1.5 (Characteristic and Fully invariant subgroups), 1.5.6(iii), More info
  • Topics in Algebra by I. N. Herstein, Page 70, Problem 9, More info
  • Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, Page 4, Section 1.1, (passing mention)More info

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