Normal versus characteristic

This survey article compares, and contrasts, the following subgroup properties: normal subgroup versus characteristic subgroup
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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.

Summary of metaproperty satisfactions

Metaproperty	Satisfied by normality?	Satisfied by characteristicity?
Auto-invariance property	Yes	Yes
Endo-invariance property	Yes	Yes
Invariance property	Yes	Yes
Transitive subgroup property	No	Yes
Strongly intersection-closed subgroup property	Yes	Yes
Strongly join-closed subgroup property	Yes	Yes
Intermediate subgroup condition	Yes	No
Transfer condition	Yes	No
Quotient-transitive subgroup property	Yes	Yes
Centralizer-closed subgroup property	Yes	Yes
Commutator-closed subgroup property	Yes	Yes

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 $H$ 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

Further information: Invariance implies strongly intersection-closed, Normality is strongly intersection-closed, Characteristicity is 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

Further information: Endo-invariance implies strongly join-closed, Normality is strongly join-closed, Characteristicity is 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

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 intermediate subgroup condition

Further information: 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

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

Further information: Potentially characteristic implies normal, Finite NPC theorem, Finite normal implies potentially characteristic, Central implies potentially characteristic, NPC conjecture

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

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

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 $K$.

Image-potentially characteristic subgroup

Further information: Finite NIPC theorem, NIPC conjecture

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

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

Further information: 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

Further information: 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 $G$.

Upper join-closedness

Normality is upper join-closed

Further information: 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 $⟨K_1,K_2⟩$. 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

Further information: 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 $⟨K_1,K_2⟩$.

Thus, there is no analogue of normalizer for characteristicity.

Corresponding notions of simplicity

Simple group

Further information: 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

Further information: 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.

Occurrence as subgroups

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.

Importance

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.