Metaproperty satisfaction analysis for right transiters of normality
This survey article looks at various subgroup properties that all behave as follows: any normal subgroup of a subgroup with property is normal in the whole group.
We discuss how these properties behave with respect to intersections, joins, centralizers, and many other operations.
The motivation behind these properties is to think of normality in terms of a function restriction expression. The standard function restriction expression for normality is:
In other words, is a normal subgroup of if any inner automorphism of restricts to an automorphism of .
However, the left side can be tightened in different ways:
- Subgroup-conjugating automorphism Automorphism
- Class-preserving automorphism Automorphism
- Normal automorphism Automorphism
The properties themselves
The four properties that are of direct interest to us are:
- Central factor: This is defined as inner automorphism inner automorphism. In other words, is a central factor of if and only if every inner automorphism of restricts to an inner automorphism of . Equivalently, .
- SCAB-subgroup: This is defined as inner automorphism subgroup-conjugating automorphism, or equivalently, subgroup-conjugating automorphism subgroup-conjugating automorphism. In other words, is a SCAB-subgroup of if any inner automorphism of restricts to an automorphism of sending subgroups to conjugate subgroups in .
- Conjugacy-closed normal subgroup: This is defined as inner automorphism class-preserving automorphism, or equivalently, as class-preserving automorphism class-preserving automorphism.
- Transitively normal subgroup: This is defined as normal automorphism normal automorphism, or equivalently, as inner automorphism normal automorphism. is transitively normal in if every normal subgroup of is normal in .
Some other properties
There are other properties that come up.
Some properties that are stronger than the property of being a central factor:
- Direct factor: This is a subgroup that is normal and has a normal complement.
- Central subgroup: This is a subgroup contained in the center.
- Cocentral subgroup: This is a subgroup whose product with the center is the whole group.
Some properties that are obtained by stripping the normal from some of the properties earlier:
- Subset-conjugacy-closed subgroup: A subgroup of a group is subset-conjugacy-closed in if, for any subsets such that there exists with , there exists such that for all . A subgroup is a central factor if and only if it is both subset-conjugacy-closed and normal.
- Conjugacy-closed subgroup: A subgroup of a group is conjugacy-closed if, whenever two elements of are conjugate in , they are conjugate in . A conjugacy-closed normal subgroup (seen earlier) is a subgroup that is both conjugacy-closed and normal.
- Central factor of normalizer: A subgroup that is a central factor in its normalizer.
Finally, one more property that we shall be frequently looking at is the property of being a retract. A retract is a subgroup that possesses a normal complement; equivalently, it is the image of an idempotent endomorphism of the whole group.
All the properties discussed here are transitive, with the exception of the property of being a central factor of normalizer. The explanations are provided below:
- The properties central factor, conjugacy-closed normal subgroup, SCAB-subgroup, and transitively normal subgroup are all transitive. This is easily seem from the fact that they are balanced with respect to the function restriction formalism: they all have function restriction expressions with the left and right side equal. For full proof, refer: Balanced implies transitive
- Direct factor is transitive
- Cocentrality is transitive
- Any subgroup of a central subgroup is central, so centrality is transitive.
- Subset-conjugacy-closedness is transitive, conjugacy-closedness is transitive
- Retract is transitive
The property of being a central factor of normalizer is not transitive. Further information: Central factor of normalizer is not transitive
Intermediate subgroup condition
All the properties given here satisfy the intermediate subgroup condition. In other words, for any of these properties , if and satisfies property in , then satisfies property in .
All the properties described here satisfy the intermediate subgroup condition:
- The properties central factor, conjugacy-closed normal subgroup, SCAB-subgroup, and transitively normal subgroup all satisfy the intermediate subgroup condition. This is because they all have function restriction expressions where the property on the left is inner automorphism. In particular, they are left-inner. For full proof, refer: Left-inner implies intermediate subgroup condition
- Direct factor satisfies intermediate subgroup condition
- Cocentrality satisfies intermediate subgroup condition
- Centrality satisfies intermediate subgroup condition
- Subset-conjugacy-closedness satisfies intermediate subgroup condition, Conjugacy-closedness satisfies intermediate subgroup condition
- Retract satisfies intermediate subgroup condition
Most of the properties discussed here do not satisfy the transfer condition. In other words, we can have a situation where are subgroups of with satisfying property in , but does not satisfy property in .
Here is the lone property that satisfies transfer condition:
Here are the many that do not satisfy transfer condition:
- Direct factor does not satisfy transfer condition, Cocentrality does not satisfy transfer condition
- Central factor does not satisfy transfer condition, transitive normality does not satisfy transfer condition, SCAB does not satisfy transfer condition (all rely on essentially the same example).
- Conjugacy-closedness does not satisfy transfer condition, Subset-conjugacy-closedness does not satisfy transfer condition, Retract does not satisfy transfer condition
All of the properties discussed here satisfy the image condition. In other words, the image, under a surjective homomorphism, of a subgroup satisfying any of these properties, also satisfies that property:
- Central factor satisfies image condition
- SCAB satisfies image condition
- Transitive normality satisfies image condition
- Direct factor satisfies image condition
- Centrality satisfies image condition
- Cocentrality satisfies image condition
- Conjugacy-closedness satisfies image condition
- Subset-conjugacy-closedness satisfies image condition
- Retract satisfies image condition
Most of the properties discussed here are not closed under taking finite intersections. The exception is the property of being a central subgroup, which is closed under taking a finite (nonzero) number of intersections.
- Direct factor is not finite-intersection-closed
- Central factor is not finite-intersection-closed, Transitive normality is not finite-intersection-closed, SCAB is not finite-intersection-closed, Conjugacy-closed normality is not finite-intersection-closed
- Conjugacy-closedness is not finite-intersection-closed, Subset-conjugacy-closedness is not finite-intersection-closed
- Cocentrality is not finite-intersection-closed
Moreover, intersections of subgroups of this kind can be very bad in general.
- Central subgroup is closed under joins, because central subgroups are precisely subgroups contained in the center.
- Cocentral subgroup is closed under (nonempty) joins, because cocentrality is upward-closed: any subgroup containing a cocentral subgroup is cocentral.
- Direct factor is not finite-join-closed
- Central factor is not finite-join-closed, SCAB is not finite-join-closed, Conjugacy-closed normality is not finite-join-closed, Transitive normality is not finite-join-closed
- Conjugacy-closedness is not finite-join-closed, Retract is not finite-join-closed
However, there are interesting relations between the properties. For instance, define a subgroup to be a join-transitively central factor if its join with any central factor is a central factor. Then, any central subgroup, cocentral subgroup, or direct factor is a join-transitively central factor. In other words, a join of two central factors of which one is a direct factor, a central subgroup, or a cocentral subgroup, is again a central factor. For full proof, refer: Direct factor implies join-transitively central factor, Central implies join-transitively central factor
On the other hand, if we define a join-transitively transitively normal subgroup as a subgroup whose join with any transitively normal subgroup is transitively normal, neither direct factors nor central subgroups are necessarily join-transitively transitively normal. Cocentral subgroups are join-transitively transitively normal.
A centralizer-closed subgroup property is a subgroup property such that the centralizer of any subgroup satisfying the property also satisfies the property. Some of the subgroup properties here are centralizer-closed, and others are not.
- Direct factor is not centralizer-closed
- Central factor is centralizer-closed
- Transitive normality is not centralizer-closed
A quotient-transitive subgroup property is a subgroup property such that whenever are such that is normal in , satisfies in and satisfies in , then satisfies in . Some of the subgroup properties here are quotient-transitive, and others are not: