Normal versus characteristic

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

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

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

Every characteristic subgroup is 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
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.

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
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
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
We have observed above that every characteristic subgroup of a normal subgroup is normal. A deeper question is: if a subgroup $$H$$ of a group $$G$$ is such that whenever $$G$$ is normal in some bigger group $$K$$, so is $$H$$, must $$H$$ be characteristic in $$G$$? 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).

Common role as invariance properties
Both normality and characteristicity are invariance properties, as can be seen from the function restriction expressions:

Normal = Inner automorphism $$\to$$ Function

In other words, normality can be described as invariance under a collection of functions, namely the inner automorphisms.

Characteristic = Automorphism $$\to$$ Function

In other words, characteristicity can be described as invariance under a collection of functions, namely all the automorphisms.

In fact, in both cases, the collection of functions are endomorphisms, hence both normality and characteristicity are endo-invariance properties. This leads to very similar behavior.

Both are strongly intersection-closed
If $$p$$ is an invariance property, then $$p$$ is strongly intersection-closed: an arbitrary (possibly empty) intersection of subgroups with property $$p$$ also has property $$p$$. In particular, an arbitrary intersection of normal subgroups is normal and an arbitrary intersection of characteristic subgroups is characteristic.

This leads to the following parallel notions:


 * Normal closure of a subgroup: Intersection of all normal subgroups containing the given subgroup, which is hence also normal, and hence the smallest normal subgroup containing the given subgroup
 * Characteristic closure of a subgroup: Intersection of all characteristic subgroups containing the given subgroup, which is hence also characteristic, and the smallest characteristic subgroup containing the given subgroup

Because every characteristic subgroup is normal, the characteristic closure in general contains the normal closure. They're equal if and only if the normal closure is characteristic, in which case the subgroup is termed a closure-characteristic subgroup.

Normality, however, satisfies some stronger versions of being intersection-closed, namely normality is strongly UL-intersection-closed. This states that if $$H_i \le K_i \le G$$ are such that each $$H_i$$ is normal in $$K_i$$, then the intersection of the $$H_i$$s is normal in the intersection of the $$K_i$$s. The corresponding statement is not true for characteristicity.

Both are strongly join-closed
If $$p$$ is an endo-invariance property, then $$p$$ is strongly join-closed: an arbitrary (possibly empty) join of subgroups with property $$p$$ also has property $$p$$. In particular, an arbitrary join of normal subgroups is normal and an arbitrary join of characteristic subgroups is characteristic.

This leads to the following parallel notions:


 * Normal core of a subgroup: Join of all normal subgroups inside the given subgroup, which is hence also normal, and hence the largest normal subgroup inside the given subgroup
 * Characteristic core of a subgroup: Join of all characteristic subgroups inside the given subgroup, which is hence also characteristic, and hence the largest characteristic subgroup inside the given subgroup.

Because every characteristic subgroup is normal, the characteristic core is contained inside the normal core. They're equal if and only if the normal core is characteristic, in which case the subgroup is termed a core-characteristic subgroup.

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
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 intermediate 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
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
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. Thus, the property of being potentially characteristic is somewhere between the properties of characteristicity and normality.

It is unknown whether every normal subgroup is potentially characteristic. However, there are partial results, such as the finite NPC theorem, and some other facts proved using amalgam-characteristic subgroups.

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

Characteristicity does not satisfy image condition
If $$\varphi:G \to K$$ is a surjective homomorphism and $$N$$ is a characteristic subgroup of $$G$$, then $$\varphi(N)$$ is not necessarily characteristic in $$K$$.

Image-potentially characteristic subgroup
A subgroup $$H$$ of a group $$K$$ is termed an image-potentially characteristic subgroup if there exists a group $$G$$, a surjective homomorphism $$\varphi:G \to K$$, and a characteristic subgroup $$L$$ of $$G$$ such that $$\varphi(L) = H$$.

It turns out that in a finite group, and in a number of other cases, every normal subgroup is image-potentially characteristic.

Quotient-transitivity
Both normality and characteristicity are quotient-transitive.

Normality is quotient-transitive
If $$H \le K \le G$$ are such that $$H$$ is normal in $$G$$ and $$K/H$$ is normal in $$G/H$$, then $$K$$ is normal in $$G$$.

Characteristicity is quotient-transitive
If $$H \le K \le G$$ are such that $$H$$ is characteristic in $$G$$ and $$K/H$$ is characteristic in $$G/H$$, then $$K$$ is characteristic in $$G$$.

Normality is upper join-closed
If $$H \le G$$ and $$K_1, K_2$$ are subgroups of $$G$$ containing $$H$$, such that $$H$$ is normal in both $$K_1$$ and $$K_2$$, then $$H$$ is normal in the join of subgroups $$\langle K_1, K_2 \rangle$$. An analogous result holds for arbitrary joins.

This, along with the fact that normality satisfies intermediate subgroup condition, allows us to talk of the normalizer of any subgroup: the largest subgroup within which a given subgroup is normal. In other words, normality is an izable subgroup property.

Characteristicity is not upper join-closed
We can have a subgroup $$H$$ of $$G$$ such that $$H$$ is characteristic in two intermediate subgroups $$K_1, K_2$$ but $$H$$ is not characteristic in the join $$\langle K_1, K_2 \rangle$$.

Thus, there is no analogue of normalizer for characteristicity.

Simple group
A simple group is a nontrivial group that contains no proper nontrivial normal subgroups. Simple groups play an important role as building blocks of all groups. For instance, every finite group has a composition series: a descending chain of subgroups, each normal in its predecessor, with the quotients being simple groups. Simple groups are important throughout mathematics.

Characteristically simple group
A characteristically simple group is a nontrivial group that contains no proper nontrivial characteristic subgroups. Characteristically simple groups are not too important in the study of groups. Moreover, their structure is largely governed by the structure of simple groups. For instance, any finite characteristically simple group is a direct product of simple groups. There do exist other infinite characteristically simple groups; for instance, the additive group of a field is characteristically simple.

Normality: more frequent
The condition of being a normal subgroup is a fairly weak one. For instance:


 * Every subgroup of an Abelian group is normal
 * Every subgroup inside the center, every subgroup inside a cyclic normal subgroup, and any subgroup containing the commutator subgroup, is normal
 * Every direct factor is normal

Thus, properties that assert the existence of normal subgroups of certain types, are not usually very restrictive.

Characteristicity: more rare
The condition of being a characteristic subgroup is a fairly strong one. It is not true that every subgroup in an Abelian group is characteristic. In fact, every subgroup being characteristic is an indication that the given group is a cyclic group. Similarly there are no results analogous to the statement that every subgroup inside the center or every subgroup containing the commutator subgroup is characteristic. In other words, being a hereditarily characteristic subgroup or being an upward-closed characteristic subgroup are very strong constraints.

Thus, postulating the existence of a characteristic subgroup with certain properties can pose strong structural restrictions (for instance, the existence of a small characteristic subgroup, the existence of a large characteristic subgroup). Conversely, a result that establishes the existence of a certain kind of characteristic subgroup in a general setting, is an extremely strong and powerful result. One such example is the critical subgroup theorem.

Normal subgroups
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.