Normal versus characteristic

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Introduction

This article is about a relation, the similarity and contrast, between two of the most important subgroup properties: normality and characteristicity.

Definitions

Normal subgroup

Further information: normal subgroup

A subgroup of a group is said to be normal if it satisfies the following equivalent conditions:

• It is invariant under all inner automorphisms. Thus, normality is the invariance property with respect to the property of an automorphism being inner. This definition also motivates the term invariant subgroup for normal subgroup (which was used earlier).
• It is the kernel of a homomorphism from the group.
• It equals each of its conjugates in the whole group. This definition also motivates the term self-conjugate subgroup for normal subgroup (which was used earlier).
• Its left cosets are the same as its right cosets (that is, it commutes with every element of the group)

Characteristic subgroup

A subgroup of a group is termed characteristic if it satisfies the following equivalent conditions:

• Every automorphism of the whole group takes the subgroup to within itself
• Every automorphism of the group restricts to an endomorphism of the subgroup
• Every automorphism of the group restricts to an automorphism of the subgroup

Formalisms

Function restriction formalism

In the function restriction formalism, we write normality as:

Inner automorphism $\to$ Function

Which means that every inner automorphism of the whole group must restrict to a function on the subgroup.

In contrast, characteristicity is written as:

Automorphism $\to$ Function

Which means that every automorphism (regardless of whether or not it is inner) of the whole group must restrict to a function on the subgroup.

We have some alternative expressions for normality, characteristicity:

• Normal = Inner automorphism $\to$ Endomorphism = Inner automorphism $\to$Automorphism
• Characteristic = Automorphism $\to$ Endomorphism = Automorphism $\to$ Automorphism

Implication relations

Every characteristic subgroup is normal

Further information: characteristic implies normal

Every characteristic subgroup of a group is normal. This is because characteristicity requires invariance under all automorphisms, which in turn guarantees invariance under inner automorphisms.

Every normal subgroup need not be characteristic

Further information: Normal not implies characteristic

A normal subgroup of a group need not be characteristic. For instance, any nontrivial group $G$ is normal but not characteristic in $G\times G$.

A subgroup property that, along with normality, implies characteristicity, is termed a normal-to-characteristic subgroup property.

Transitivity and transiters

Normality is not transitive

Further information: Normality is not transitive

Probably one of the biggest defects of normality is that it is not a transitive subgroup property. In other words, a normal subgroup of a normal subgroup need not be normal.

Characteristicity is transitive

Further information: Characteristicity is transitive

Characteristicity is a transitive subgroup property. In other words, a characteristic subgroup of a characteristic subgroup is characteristic. The proof follows from the function restriction formal expression for characteristicity given below, which shows that characteristicity is a balanced subgroup property:

Automorphism $\to$ Automorphism

Characteristic of normal

Further information: characteristic of normal implies normal

In order to remedy the lack of transitivity of normality, we are interested in looking at situations where $H\le K$ is a subgroup such that whenever $K$ is normal in $G$, then $H$ is also normal in $G$.

It turns out that if $H$ is characteristic in $K$ and $K$ is normal in $G$, then $H$ is normal in $G$ This follows form the function restriction formal expressions:

• Characteristic = Automorphism $\to$ Automorphism
• Normal = Inner automorphism $\to$ Automorphism

Characteristicity is the left transiter

Further information: Left transiter of normal is characteristic

We have observed above that every characteristic subgroup of a normal subgroup is normal. A deeper question is: if a subgroup of a group is such that whenever the whole group is normal in some bigger group, so is the subgroup, must the subgroup be characteristic? In other words, is characteristicity precisely the left transiter of normality?

The answer is yes. The proof of this relies on the fact that every group can be embedded as a normal fully normalized subgroup of some group (for instance, of its holomorph).

Intermediate subgroup condition

We saw that characteristicity plays the role of remedying the lack of transitivity of normality. A similar role is played by normality in remedying a certain subgroup metaproperty that characteristicity does not satisfy: the intermediate subgroup condition.

Normality satisfies intermediate subgroup condition

Further information: Normality satisfies intermediate subgroup condition

If $H$ is a normal subgroup of $G$, and $K$ is an intermediate subgroup containing $H$, then $H$ is normal in $K$. In other words, normality satisfies the intermediate subgroup condition.

Characteristicity does not satisfy imtermediate subgroup condition

If $H$ is characteristic in $G$, and $K$ is an intermediate subgroup, $H$ need not be characteristic in $K$. In other words, characteristicity does not satisfy the intermediate subgroup condition.

Potentially characteristic subgroup

Further information: Potentially characteristic subgroup

The potentially operator takes as input a subgroup property $p$ and outputs the property $q$ such that:

$H$ has property $q$ in $K$ if there exists a group $G$ containing $K$ such that $H$ has property $p$ in $G$.

Putting $p$ to be characteristicity, we get the notion of potentially characteristic subgroup. In plainspeak, $H$ is potentially characteristic in $K$ if there exists a group $G$ containing $K$ such that $H$ is characteristic in $G$.

Potentially characteristic versus normal

Note that if a subgroup property satisfies the intermediate subgroup condition, it is invariant under the potentially operator. Thus, any potentially characteristic subgroup is potentially normal, and hence normal.

What is interesting is the converse question: is every normal subgroup potentially characteristic?

Image condition

Normality rectifies another property that characteristicity fails to have. Under a surjective homomorphism the image of a normal subgroup is normal, but the same is not true of characteristic subgroups.

Normality satisfies image condition

Further information: Normality satisfies image condition

If $\phi :G\to K$ is a surjective homomorphism and $N$ is a normal subgroup of $G$, then $\phi (N)$ is normal in $K$.

Characteristicity does not satisfy image condition

Further information: Characteristicity does not satisfy image condition, Characteristicity is quotient-transitive

If $\phi :G\to K$ is a surjective homomorphism and $N$ is a characteristic subgroup of $G$, then $\phi (N)$ is not necessarily characteristic in $K$. A weaker version is, however, true. If the kernel of $\phi$ is also characteristic, then the image of $N$ is characteristic.

Occurrence

Normal subgroups

Further information: Ubiquity of normality

Normal subgroups are important not only in group theory, but in practically any situation in which groups arise. The reason for this is, roughly, that the inner automorphisms of a group play a role even in situations where we are interested in the group acting on some structure and not as a group in itself.

Characteristic subgroups

These are important largely within group theory, when trying to fix the abstract structure of a group. Characteristic subgroups do come up somewhat in geometric group theory and linear representation theory, but their role is more on the lines of a guest appearance.