Characteristic of normal implies normal

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 $\leq$ Normal

Here, $*$ denotes the composition operator.

Verbal statement

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

Statement with symbols

Let $H\leq K\leq G$ 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 $K\leq G$ 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

• Characteristicity is transitive: A characteristic subgroup of a characteristic subgroup is characteristic.
• Left transiter of normal is characteristic: Characteristicity is the weakest, or most general property, for which the above statement is true. This is made precise in the statement that characteristicity is the left transiter for normality.
• Automorph-permutable of normal implies conjugate-permutable: This statement has many corollaries; for instance, 2-subnormal implies conjugate-permutable

Analogues

• Derivation-invariant subring of ideal implies ideal: This is the analogous statement for Lie rings. Here, derivations play the role of automorphisms, Lie subrings play the role of subgroups, ideals play the role of normal subgroups, inner derivations play the role of inner automorphisms, and derivation-invariant subrings play the role of characteristic subgroups.

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:

• Hands-on definition: A subgroup $H$ of a group $G$ is termed a characteristic subgroup, if for any automorphism $\sigma$ of $G$, we have $\sigma (H)=H$.
• Definition using function restriction expression: We can write characteristicity as the balanced subgroup property with respect to automorphisms:

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 $g\in G$:

$c_{g}(H)\leq H$

or more explicitly:

$gHg^{-1}\leq H$

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 $c_{g}(H)\leq H$ as well as $c_{g^{-1}}(H)\leq H$). This equivalence of ideas is crucial to the proof.

• Definition using function restriction expression: We can write normality as the invariance property with respect to inner automorphisms:

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

• Composition rule for function restriction: This is used for the proof using function restriction expressions.

Proof

Hands-on proof

Given: groups $H\leq K\leq G$ such that $H$ is characteristic in $K$ and $K$ is normal in $G$.

To Prove: For any $g\in G$, the map $c_{g}:x\mapsto gxg^{-1}$ takes $H$ to within itself.

Proof:

1. Since $K\triangleleft G$, $c_{g}(x)\in K$ for every $x\in K$. Thus, $c_{g}$ restricts to a function from $K$ to $K$. Since this function arises by restricting an automorphism of $G$, it is an endomorphism of $K$ (i.e., a homomorphism from $K$ to itself).
2. By similar reasoning, $c_{g^{-1}}$ also restricts to an endomorphism of $K$.
3. Since $c_{g^{-1}}\circ c_{g}$ and $c_{g}\circ c_{g^{-1}}$ are both the identity map on $G$, and $K$ is invariant under both, $c_{g}$ and $c_{g^{-1}}$ are two-sided inverses of each other as maps from $K$ to itself. Thus, the restriction of $c_{g}$ to $K$ is actually an invertible endomorphism, viz., an automorphism of $K$. Call this automorphism $\sigma$.
4. Since $H$ is characteristic in $K$, $\sigma$ takes $H$ to within itself. But since $\sigma$ is the restriction of $c_{g}$ to $K$ in the first place, we conclude that $c_{g}$ in fact takes $H$ to itself. This completes the proof.

Using function restriction expressions

In terms of the function restriction formalism:

• The following is a function restriction expression for the subgroup property of normality:

Inner automorphism $\to$ Automorphism

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

• The following is a function restriction expression for the subgroup property of characteristicity:

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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