Potentially characteristic subgroups characterization problem
This article describes an open problem in the following area of/related to group theory: group theory
Contents
Statement
Main versions
Given a group and group property satisfied by , the goal of the problem is to characterize, for each of the properties given below, all the subgroups satisfying that property:
Notion | Meaning |
---|---|
characteristic-potentially characteristic subgroup relative to | A subgroup of for which there exists a group containing and satisfying such that both and are characteristic subgroups of . |
normal-potentially characteristic subgroup relative to | A subgroup of for which there exists a group containing and satisfying such that is characteristic in and is normal in . |
normal-potentially relatively characteristic subgroup relative to | A subgroup of for which there exists a group containing and satisfying such that is normal in and is invariant under all the automorphisms of that extend to automorphisms of . |
potentially characteristic subgroup relative to | A subgroup of for which there exists a group containing and satisfying such that is characteristic in . |
image-potentially characteristic subgroup | A subgroup of such that there is a surjective homomorphism such that satisfies and a characteristic subgroup of such that . |
semi-strongly image-potentially characteristic subgroup relative to | A subgroup of such that there is a surjective homomorphism from a group satisfying such that is characteristic in . |
strongly image-potentially characteristic subgroup relative to | A subgroup of such that there is a surjective homomorphism from a group satisfying such that the kernel of and are both characteristic in . |
retract-potentially characteristic subgroup relative to | A subgroup of for which there exists a group containing as a retract and satisfying such that is normal in and is invariant under all the automorphisms of that extend to automorphisms of . |
They all lie between characteristic subgroup and normal subgroup (within a group of the respective property).
More general formulations
We have the following more general operators:
Operator | Inputs to the operator | Meaning |
---|---|---|
potentially operator | subgroup property | A subgroup of a group satisfies the property potentially if there exists a group containing such that satisfies the property in . |
group property-conditionally potentially operator | group property , subgroup property | A subgroup of a group satisfying property is said to be potentially conditional to if there exists a group containing and satisfying such that has property in . |
image-potentially operator | subgroup property | A subgroup of a group satisfies the property image-potentially if there exists a group with a surjective homomorphism and a subgroup of such that and satisfies in . |
group property-conditionally image-potentially operator | group property , subgroup property | A subgroup of a group satisfies the property image-potentially conditional to if there exists a group satisfying with a surjective homomorphism and a subgroup of such that and satisfies in . |
- group property-conditionally potentially verbal subgroup
- group property-conditionally potentially fully invariant subgroup
- group property-conditionally image-potentially fully invariant subgroup
Known best results for equality with normality
The major results are the NPC theorem, NIPC theorem, finite NPC theorem, and finite NIPC theorem which show that for all groups as well as for finite groups, potentially characteristic subgroups and strongly image-potentially characteristic subgroups are normal.
Contrary results are results such as the fact that normal not implies normal-potentially characteristic, the proof of which works both for all groups and for finite groups. Also of note are contrary results for infinite abelian groups.
The most surprising contrary result is the following: every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it. This gives plenty of examples of subgroups that are not finite-p-potentially characteristic subgroups.
group property | characteristic-potentially characteristic | normal-potentially characteristic | potentially characteristic | image-potentially characteristic | semi-strongly image-potentially characteristic | strongly image-potentially characteristic | retract-potentially characteristic |
---|---|---|---|---|---|---|---|
all groups | no | no | yes | yes | yes | yes | yes |
finite group | no (same as above) | no (same as above) | yes | yes | yes | yes | yes |
abelian group | no | no | no | no | no | no | no |
finite abelian group | no | yes | yes | yes | yes | no | |
group of prime power order | no | no | no | ? | ? | ? | ? |
More:
group property | potentially fully invariant | normal-potentially fully invariant | image-potentially fully invariant | potentially verbal | normal-potentially verbal | verbal-potentially verbal |
---|---|---|---|---|---|---|
all groups | no | no | no | no | no | no |
finite group | no (same as above) | no | ? | no | no | no |
finite abelian group | yes | yes | ? | yes | yes | no |
abelian group | no | no | no | no | no | no |
Normal-potentially characteristic subgroups: elaboration
The property of being a normal-potentially characteristic subgroup is strictly between characteristicity and normality. The following results are known:
- Normal-potentially characteristic implies normal-extensible automorphism-invariant, normal not implies normal-extensible automorphism-invariant
- Normal-potentially characteristic not implies characteristic
- Normal not implies normal-potentially characteristic
Further information: potentially normal-subhomomorph-containing equals normal
The following properties, all closely related to characteristicity, give normality when we apply the potentially operator:
- strictly characteristic subgroup: invariant under all surjective endomorphisms.
- normal-homomorph-containing subgroup: contains any homomorphic image that is normal in the whole group.
- normal-subhomomorph-containing subgroup: contains any homomorphic image of a subgroup, that is normal in the whole group.
- Normal not implies potentially fully invariant
- Normal not implies potentially verbal
- Normal not implies image-potentially fully invariant
General construction ideas for collapse to normality
The wreath product idea
This idea involves taking a wreath product of a coprime group with acting via the regular action of . This is used to prove the NPC theorem, NIPC theorem, finite NPC theorem, finite NIPC theorem. The same construction also shows that normality equals the property obtained by applying the potentially operator to strictly characteristic subgroup, normal-homomorph-containing subgroup, and normal-subhomomorph-containing subgroup. Further information: potentially normal-subhomomorph-containing equals normal
The amalgam idea
This idea involves taking the amalgam of with itself with as the normal subgroup. If becomes characteristic in the amalgam, it is termed an amalgam-characteristic subgroup. It turns out that finite normal implies amalgam-characteristic, periodic normal implies amalgam-characteristic, central implies amalgam-characteristic, and normal subgroup contained in hypercenter is amalgam-characteristic. However, characteristic not implies amalgam-characteristic and direct factor not implies amalgam-characteristic.
The central product idea to get a verbal or fully invariant subgroup
This idea works in a limited range of cases: try to get a huge subgroup containing and take a central product or some variant thereof of and to get a group . Some of the results using this approach are:
- Central implies finite-pi-potentially verbal in finite
- Cyclic normal implies finite-pi-potentially verbal in finite
- Homocyclic normal implies finite-pi-potentially fully invariant in finite
- Abelian normal subgroup of finite group with induced quotient action by power automorphisms is finite-pi-potentially verbal
- Central and additive group of a commutative unital ring implies potentially iterated commutator subgroup in solvable group
General ideas for proving non-collapse to normality
Rigidity of structure enforced by few outer automorphisms
In the proof that normal not implies normal-potentially characteristic, we use the fact that if the ambient group is normal in some bigger group , then acts on by automorphisms. We next try to determined the image of in , which is a subgroup containing . Some of the results we obtain this way are: centerless and maximal in automorphism group implies every automorphism is normal-extensible, every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible.
Injective and projective modules
In the case of abelian groups, we use the fact that divisible abelian groups are injective modules in the sense that they are always direct summands in any group in which they are contained. This allows us to show that all automorphisms of such gorups are abelian-extensible automorphisms and a non-characteristic subgroup of such a group cannot be an abelian-potentially characteristic subgroup.
Similarly, free abelian groups are projective modules, and we can use this to show that any non-characteristic subgroup of a free abelian group is not an abelian-image-potentially characteristic subgroup, because all automorphisms of free abelian groups are abelian-quotient-pullbackable automorphisms.
Analogues of injective and projective modules for non-abelian groups
For non-abelian groups, the analogous notion to injective module is that of complete group. A complete normal subgroup is a direct factor. We use complete groups to show that normal not implies potentially fully invariant.
The analogous notion to projective module is that of free group. We use free groups to show that normal not implies image-potentially fully invariant.
Working with groups of prime power order
Within a group of prime power order, we can use techniques of rigidity of structure and few automorphisms for certain groups to obtain results such as every finite p-group is a subgroup of a finite p-group that is not characteristic in any finite p-group properly containing it. This was proved in a paper by Bettina Wilkens.