Potentially characteristic subgroups characterization problem

This article describes an open problem in the following area of/related to group theory: group theory

Statement

Main versions

Given a group $G$ and group property $\alpha$ satisfied by $G$, 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 $\alpha$	A subgroup $H$ of $G$ for which there exists a group $K$ containing $G$ and satisfying $\alpha$ such that both $H$ and $G$ are characteristic subgroups of $G$.
normal-potentially characteristic subgroup relative to $\alpha$	A subgroup $H$ of $G$ for which there exists a group $K$ containing $G$ and satisfying $\alpha$ such that $H$ is characteristic in $K$ and $G$ is normal in $K$.
normal-potentially relatively characteristic subgroup relative to $\alpha$	A subgroup $H$ of $G$ for which there exists a group $K$ containing $G$ and satisfying $\alpha$ such that $G$ is normal in $K$ and $H$ is invariant under all the automorphisms of $G$ that extend to automorphisms of $K$.
potentially characteristic subgroup relative to $\alpha$	A subgroup $H$ of $G$ for which there exists a group $K$ containing $G$ and satisfying $\alpha$ such that $H$ is characteristic in $K$.
image-potentially characteristic subgroup	A subgroup $H$ of $G$ such that there is a surjective homomorphism $\rho:K \to G$ such that $K$ satisfies $\alpha$ and a characteristic subgroup $L$ of $K$ such that $\rho(L) = H$.
semi-strongly image-potentially characteristic subgroup relative to $\alpha$	A subgroup $H$ of $G$ such that there is a surjective homomorphism $\rho:K \to G$ from a group $K$ satisfying $\alpha$ such that $\rho^{-1}(H)$ is characteristic in $K$.
strongly image-potentially characteristic subgroup relative to $\alpha$	A subgroup $H$ of $G$ such that there is a surjective homomorphism $\rho:K \to G$ from a group $K$ satisfying $\alpha$ such that the kernel of $\rho$ and $\rho^{-1}(H)$ are both characteristic in $K$.
retract-potentially characteristic subgroup relative to $\alpha$	A subgroup $H$ of $G$ for which there exists a group $K$ containing $G$ as a retract and satisfying $\alpha$ such that $G$ is normal in $K$ and $H$ is invariant under all the automorphisms of $G$ that extend to automorphisms of $K$.

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 $\beta$	A subgroup $H$ of a group $G$ satisfies the property potentially $\beta$ if there exists a group $K$ containing $G$ such that $H$ satisfies the property $\beta$ in $K$.
group property-conditionally potentially operator	group property $\alpha$, subgroup property $\beta$	A subgroup $H$ of a group $G$ satisfying property $\alpha$ is said to be potentially $\beta$ conditional to $\alpha$ if there exists a group $K$ containing $G$ and satisfying $\alpha$ such that $H$ has property $\beta$ in $K$.
image-potentially operator	subgroup property $\beta$	A subgroup $H$ of a group $G$ satisfies the property image-potentially $\beta$ if there exists a group $K$ with a surjective homomorphism $\rho:K \to G$ and a subgroup $L$ of $K$ such that $\rho(L) = H$ and $L$ satisfies $\beta$ in $K$.
group property-conditionally image-potentially operator	group property $\alpha$, subgroup property $\beta$	A subgroup $H$ of a group $G$ satisfies the property image-potentially $\beta$ conditional to $\alpha$ if there exists a group $K$ satisfying $\alpha$ with a surjective homomorphism $\rho:K \to G$ and a subgroup $L$ of $K$ such that $\rho(L) = H$ and $L$ satisfies $\beta$ in $K$.

Other related cases of interest

• 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

Things related to characteristicity for which the potentially operator gives normality

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.

Potentially fully invariant subgroups and related ideas

• 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 $G$ acting via the regular action of $G/H$. 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 $G$ with itself with $H$ as the normal subgroup. If $H$ 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 $L$ containing $H$ and take a central product or some variant thereof of $L$ and $G$ to get a group $K$. 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 $G$ is normal in some bigger group $K$, then $K$ acts on $G$ by automorphisms. We next try to determined the image of $K$ in $\operatorname{Aut}(G)$, which is a subgroup containing $\operatorname{Inn}(G)$. 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.