Varying normality

This is a survey article describing variations on the following: normal subgroup
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Normality is one of the most pivotal subgroup properties. It traces its origins to the very beginnings of group theory, in fact, to even before that. Given its long history and the varied ways in which it turns up, it is natural that a large number of variations of normality have popped up in group theory.

A full list of the subgroup properties obtained by varying normality is available at:

Variations of normal subgroup

This article surveys some of the more common among the many variations of the subgroup property of normality, trying to organize them into themes and streams. There are three basic ideas behind variation:

• Emulate the strengths
• Remedy the weaknesses
• Weaken or remove the strengths

Also refer the variational charts for variations stronger than normality and variations weaker than normality.

Variations stronger than normality

Variations weaker than normality

Finding right transiters

Lack of transitivity is one of the major problems with normality (in other words, a normal subgroup of a normal subgroup need not be normal). One way of remedying this problem is to find transitive subgroup properties $q$ such that every normal subgroup of a subgroup with property $q$ is normal.

All the properties $q$ discussed below satisfy the following:

• Every normal subgroup of a subgroup with property $q$ is also normal
• The subgroup property is transitive
• The subgroup property satisfies intermediate subgroup condition

Transitively normal subgroup

Further information: transitively normal subgroup

The property of being a transitively normal subgroup is the right transiter for the subgroup property of normality. It is defined as follows:

$H$ is transitively normal in $G$ if whenever $K$ is a normal subgroup of $H$, then $K$ is also normal in $G$.

Alternatively, observe that the property of being a normal subgroup can be expressed in the function restriction formalism as:

Normal automorphism $\to$ Automorphism

This is a left tight function restriction expression, and hence the right transiter of normality is:

Normal automorphism $\to$ Normal automorphism

This is the same as the property of being transitively normal. In other words, $H$ is transitively normal in $G$ if and only if every automorphism of $G$ that preserves normal subgroups in $G$ restricts to an automorphism of $H$ that preserves normal subgroups in $H$

Conjugacy-closed normal subgroup

Further information: conjugacy-closed normal subgroup

The property of being a conjugacy-closed normal subgroup is equivalent to the property of being both normal, and conjugacy-closed. A subgroup is termed conjugacy-closed if any two elements in the subgroup that are conjugate in the whole group are also conjugate in the subgroup.

Alternatively, we can view the property of being conjugacy-closed normal as follows.

The property of being normal can be written as:

Class-preserving automorphism $\to$ Automorphism

Hence the property:

Class-preserving automorphism $\to$ Class-preserving automorphism

is stronger than the right transiter of normality. This property is precisely the same as the property of being a conjugacy-closed normal subgroup.

Central factor

Further information: central factor

The property of being a central factor is defined as follows: $H$ is a central factor of $G$ if $HC_G(H) = G$. Equivalently, observe that the subgroup property of normality can be expressed as:

Inner automorphism $\to$ Automorphism

Thus, the following property is clearly stronger than the right transiter:

Inner automorphism $\to$ Inner automorphism

This is precisely the same as the property of being a central factor. In other words, a subgroup $H$ is a central factor of a group $G$ if and only if every inner automorphism of $G$ restricts to an inner automorphism of $H$.

Direct factor

Further information: direct factor

A direct factor of a group is a subgroup that is one of the factors in an internal direct product.

We know that the subgroup property of being a direct factor satisfies the intermediate subgroup condition -- a direct factor in the whole group is also a direct factor in any intermediate subgroup. Further, a direct factor is clearly a central factor, hence it is stronger than the right transiter of normality.

Finding left transiters

Characteristic subgroup

Further information: characteristic subgroup, characteristic of normal implies normal, left transiter of normal is characteristic The property of normality can be expressed as:

Inner automorphism $\to$ Automorphism

Further, this expression is right tight for normality, hence the left transiter of normality is the property:

Automorphism $\to$ Automorphism

This is the property of being a characteristic subgroup.

Obtaining a handle on the quotient

When we have a normal subgroup, there's a natural quotient group. Two questions arise:

• What can we say about the quotient as an abstract group?
• To what extent can we realize the quotient as a subgroup?

Direct factor

Further information: direct factor

A direct factor is a normal subgroup that has a complement which is also a normal subgroup. In other words, $H$ is a direct factor in $G$ if there is a subgroup $K$ of $G$ such that $H$ and $K$ are both normal, $H \cap K$ is trivial, and $HK=G$.

Notice that if $H$ is a direct factor of $G$, the quotient group $G/H$ is isomorphic to $K$ in a natural map -- the map that sends each coset of $H$ in $G$ to the unique element of $K$ in that coset.

Complemented normal subgroup

Further information: complemented normal subgroup

A complemented normal subgroup is a normal subgroup $H$ such that there exists a subgroup $K$ of $G$ such that $H \cap K$ is trivial and $HK = G$. We no longer assume that $K$ is also normal.

The quotient group $G/H$ is isomorphic to $K$ via the map that sends each coset of $H$ to the unique element of $K$ that lies inside that coset.

This is still somewhat nice: it means that the quotient occurs as a subgroup in a sufficiently natural way.

Related notions are the notion of retract (in this setup $K$ is a retract and the quotient map from $G$ to $K$ is a retraction) and the notion of semidirect product (here $G$ is the internal semidirect product of $H$ by $K$).

Regular kernel

Further information: regular kernel

Suppose $G$ is generated by subgroups $A$ and $B$, with the property that $A$ intersects the normal closure of $B$ trivially, and $B$ intersects the normal closure of $A$ trivially. Then, the normal closure of $A$ are termed regular kernels in $G$, and the subgroups $A$ and $B$ are termed regular retracts.

For instance, a direct factor is a regular kernel as well as a regular retract.

Endomorphic kernel

Further information: endomorphic kernel

A somewhat weaker requirement than being able to find a complement to the normal subgroup is being able to find a subgroup that is isomorphic as an abstract group to the quotient.

A normal subgroup is termed an endomorphic kernel if it occurs as the kernel of an endomorphism from the group, or equivalently, if there is a subgroup of the group isomorphic as an abstract group to the quotient by this normal subgroup.

Subordination

To remedy the lack of transitivity of normality, we can take some subgroup properties that involve repeatedly taking normal subgroups starting from the whole group. This gives various notions.

2-subnormal subgroup

Further information: 2-subnormal subgroup

A subgroup is termed 2-subnormal if it is a normal subgroup of a normal subgroup of the whole group.

Subnormal subgroup

Further information: subnormal subgroup

A subgroup is said to be subnormal if there is a finite ascending chain of subgroups, starting from the subgroup, and ending at the whole group, such that each is normal in its successor.

Equivalently, it is obtained by applying the subordination operatorto the subgroup property of being normal.

Ascendant subgroup

Further information: ascendant subgroup

A subgroup is said to be ascendant in the whole group if there is a (possibly transfinite, well-ordered) ascending chain of subgroups, starting at the subgroup, and ending at the whole group, such that at each ordinal, the union of the subgroups corresponding to strictly smaller ordinals, is normal in the subgroup corresponding to that ordinal.

Descendant subgroup

Further information: Descendant subgroup

A subgroup is said to be ascendant in the whole group if there is a (possibly transfinite, well-ordered) descending chain of subgroups, starting at the whole group, and ending at the subgroup, such that at each ordinal, the union of the subgroups corresponding to strictly smaller ordinals, is normal in the subgroup corresponding to that ordinal.

Serial subgroup

Further information: serial subgroup

A subgroup is said to be serial in the whole group if there is a totally ordered collection of subgroups between the subgroup and the whole group, with the property that given any cut of this chain, the union of all subgroups to the left of the cut is normal in the intersection of all subgroups to the right of the cut.

Hypernormalized subgroup

Further information: hypernormalized subgroup

A subgroup is said to be hypernormalized if the operation of repeatedly taking normalizers in the whole group, starting from the subgroup, eventually takes us to the whole group. This eventually may be after transfinitely many steps, and hence hypernormalized subgroups need not be subnormal, but they are ascendant. For finite groups, of course, any hypernormalized subgroup is subnormal.

Normal subgroups are always hypernormalized.

Permutability

A subgroup $H$ is normal in a group $G$ if and only if for every $g \in G$, we have $Hg = gH$. Thus, normal subgroups permute with elements. We consider weakenings of this.

Permutable subgroup (or quasinormal subgroup)

Further information: Permutable subgroup

A permutable subgroup is a subgroup that permutes with every subgroup. In other words, its product with every subgroup is again a subgroup.

Automorph-permutable subgroup

Further information: Automorph-permutable subgroup

An automorph-permutable subgroup is a subgroup that permutes with all its images under automorphisms of the whole group. This property is weaker than the property of being permutable.

Conjugate-permutable subgroup

Further information: Conjugate-permutable subgroup

A conjugate-permutable subgroup is a subgroup that permutes with all its conjugate subgroups. This condition is weaker than permutability. It turns out that for finite groups, any conjugate-permutable subgroup is subnormal. For full proof, refer: Conjugate-permutable implies subnormal (finite groups)

The notion of resemblance

A normal subgroup is a subgroup such that any conjugate of it is equal to it. We can weaken this property somewhat by demanding that any conjugate of the given subgroup be very similar to it. These notions are explored in the class of variations discussed here.

All the properties discussed here are subnormal-to-normal: in particular, any subnormal subgroup satisfying any of these properties is normal. Further, all of these properties satisfy the intermediate subgroup condition: a subgroup satisfying this property in the whole group also satisfies it in any intermediate subgroup. Thus, they are all stronger than the property of being an intermediately subnormal-to-normal subgroup.

For a more detailed discussion of properties that combine with subnormality to give normality, refer: subnormal-to-normal and normal-to-characteristic.

CONVENTION WARNING: This article/section uses the right-action convention. The left and right action conventions are equally powerful and statements/reasoning here can be converted to the alternative convention (the main reason being that every group is naturally isomorphic to its opposite group via the inverse map). For more on the action conventions and switching between them, refer to switching between the left and right action conventions.

Pronormal subgroup

Further information: pronormal subgroup

A subgroup $H$ is pronormal in a group $G$, if, given any $g \in G$, $H$ and the conjugate subgroup $H^g$ are conjugate in the subgroup generated by them (and hence, they are conjugate in any intermediate subgroup containing both of them).

Pronormal subgroups are subnormal-to-normal: any pronormal subnormal subgroup is normal.

The most important examples of pronormal subgroups are Sylow subgroups (Sylow implies pronormal) and Sylow subgroups of normal subgroups (Sylow of normal implies pronormal). Many of the facts proved about Sylow subgroups generalize to pronormal subgroups.

Weakly pronormal subgroup

Further information: weakly pronormal subgroup

A subgroup $H$ is weakly pronormal in a group $G$ if given any $g \in G$, $H$ and the conjugate subgroup $H^g$ are conjugate in the subgroup $H^{\langle g \rangle}$.

Paranormal subgroup

Further information: paranormal subgroup

A subgroup $H$ is paranormal in a group $G$ if for any $g \in G$, $H$ is a contranormal subgroup inside $\langle H, H^g \rangle$. In other words, the normal closure of $H$ in $\langle H, H^g \rangle$ is $\langle H, H^g \rangle$.

Polynormal subgroup

Further information: polynormal subgroup

A subgroup $H$ is polynormal in a group $G$ if for any $g \in G$, $H$ is a contranormal subgroup inside $H^{\langle g \rangle}$.

Normal, up to finite index

These are variations of normality that are useful when studying infinite groups, and where differences of finite index are negligible.

Nearly normal subgroup

Further information: Nearly normal subgroup

A nearly normal subgroup is a subgroup that has finite index in its normal closure (the smallest normal subgroup containing it).

Almost normal subgroup

Further information: Almost normal subgroup

An almost normal subgroup is a subgroup whose normalizer has finite index in the whole group; equivalently, it has only finitely many conjugate subgroups.

Invariance under automorphisms close to inner automorphisms

Normality is the property of being invariant under inner automorphisms. There are a number of automorphism properties that are close to that of being an inner automorphism, and the invariance property corresponding to each closely resembles normality.

Invariance under extensible automorphisms

Further information: Extensible automorphism-invariant equals normal

An extensible automorphism is an automorphism of a group that extends to an automorphism for any bigger group containing it. It turns out that every extensible automorphism of a group is a subgroup-conjugating automorphism, and in particular, is a normal automorphism, so the property of being extensible automorphism-invariant equals the property of being normal.

Invariance under monomial automorphisms

Further information: Monomial automorphism-invariant subgroup

Invariance under all automorphisms

Further information: Characteristic subgroup

A subgroup which is invariant under all automorphisms of the whole group, is termed a characteristic subgroup.