This is a survey article describing variations on the following: normal subgroup
View other variational survey articles | View other survey articles about normal subgroup
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:
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
- 1 Finding right transiters
- 2 Finding left transiters
- 3 Obtaining a handle on the quotient
- 4 Subordination
- 5 Permutability
- 6 The notion of resemblance
- 7 Normal, up to finite index
- 8 Invariance under automorphisms close to inner automorphisms
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 such that every normal subgroup of a subgroup with property is normal.
All the properties discussed below satisfy the following:
- Every normal subgroup of a subgroup with property is also normal
- The subgroup property is transitive
- The subgroup property satisfies intermediate subgroup condition
Transitively normal subgroup
Further information: transitively normal subgroup
is transitively normal in if whenever is a normal subgroup of , then is also normal in .
Alternatively, observe that the property of being a normal subgroup can be expressed in the function restriction formalism as:
Normal automorphism Automorphism
This is a left tight function restriction expression, and hence the right transiter of normality is:
Normal automorphism Normal automorphism
This is the same as the property of being transitively normal. In other words, is transitively normal in if and only if every automorphism of that preserves normal subgroups in restricts to an automorphism of that preserves normal subgroups in
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 Automorphism
Hence the property:
Class-preserving automorphism 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.
Further information: central factor
The property of being a central factor is defined as follows: is a central factor of if . Equivalently, observe that the subgroup property of normality can be expressed as:
Inner automorphism Automorphism
Thus, the following property is clearly stronger than the right transiter:
Inner automorphism Inner automorphism
This is precisely the same as the property of being a central factor. In other words, a subgroup is a central factor of a group if and only if every inner automorphism of restricts to an inner automorphism of .
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
Inner automorphism 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?
Further information: direct factor
A direct factor is a normal subgroup that has a complement which is also a normal subgroup. In other words, is a direct factor in if there is a subgroup of such that and are both normal, is trivial, and .
Notice that if is a direct factor of , the quotient group is isomorphic to in a natural map -- the map that sends each coset of in to the unique element of in that coset.
Complemented normal subgroup
Further information: complemented normal subgroup
A complemented normal subgroup is a normal subgroup such that there exists a subgroup of such that is trivial and . We no longer assume that is also normal.
The quotient group is isomorphic to via the map that sends each coset of to the unique element of 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 is a retract and the quotient map from to is a retraction) and the notion of semidirect product (here is the internal semidirect product of by ).
Further information: regular kernel
Suppose is generated by subgroups and , with the property that intersects the normal closure of trivially, and intersects the normal closure of trivially. Then, the normal closure of are termed regular kernels in , and the subgroups and are termed regular retracts.
For instance, a direct factor is a regular kernel as well as a regular retract.
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.
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.
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.
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.
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.
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.
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.
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.
A subgroup is normal in a group if and only if for every , we have . Thus, normal subgroups permute with elements. We consider weakenings of this.
Permutable subgroup (or quasinormal subgroup)
Further information: 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.
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.
Further information: pronormal subgroup
A subgroup is pronormal in a group , if, given any , and the conjugate subgroup 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 is weakly pronormal in a group if given any , and the conjugate subgroup are conjugate in the subgroup .
Further information: paranormal subgroup
A subgroup is paranormal in a group if for any , is a contranormal subgroup inside . In other words, the normal closure of in is .
Further information: polynormal subgroup
A subgroup is polynormal in a group if for any , is a contranormal subgroup inside .
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
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.