# Normal versus characteristic

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

View other subgroup property comparison survey articles View all comparison survey articles View survey articles related to normal subgroup View survey articles related to 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 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 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 Endomorphism = Inner automorphism Automorphism
- Characteristic = Automorphism Endomorphism = Automorphism 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 is normal but not characteristic in .

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 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 is a subgroup such that whenever is normal in , then is also normal in .

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

- Characteristic = Automorphism Automorphism
- Normal = Inner automorphism 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 is normal in some bigger group , so is , must be characteristic in ? 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 Function

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

Characteristic = Automorphism 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 is an invariance property, then is strongly intersection-closed: an arbitrary (possibly empty) intersection of subgroups with property also has property . 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 are such that each is normal in , then the intersection of the s is normal in the intersection of the 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 is an endo-invariance property, then is strongly join-closed: an arbitrary (possibly empty) join of subgroups with property also has property . 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 is a normal subgroup of , and is an intermediate subgroup containing , then is normal in . 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 is characteristic in , and is an intermediate subgroup, need not be characteristic in . 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 and outputs the property such that:

has property in if there exists a group containing such that has property in .

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

### 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 is a surjective homomorphism and is a normal subgroup of , then is normal in .

### Characteristicity does not satisfy image condition

`Further information: Characteristicity does not satisfy image condition`

If is a surjective homomorphism and is a characteristic subgroup of , then is not necessarily characteristic in .

### Image-potentially characteristic subgroup

`Further information: Finite NIPC theorem, NIPC conjecture`

A subgroup of a group is termed an image-potentially characteristic subgroup if there exists a group , a surjective homomorphism , and a characteristic subgroup of such that .

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 are such that is normal in and is normal in , then is normal in .

### Characteristicity is quotient-transitive

`Further information: Characteristicity is quotient-transitive`

If are such that is characteristic in and is characteristic in , then is characteristic in .

## Upper join-closedness

### Normality is upper join-closed

`Further information: Normality is upper join-closed`

If and are subgroups of containing , such that is normal in both and , then is normal in the join of subgroups . 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 of such that is characteristic in two intermediate subgroups but is not characteristic in the join .

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.