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:

characteristic-potentially characteristic subgroup relative to $\alpha$ : Find all subgroups $H$ 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 $\alpha$ : Find all subgroups $H$ 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 $\alpha$ : Find all subgroups $H$ 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 $\alpha$ : Find all subgroups $H$ 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: Find all subgroups $H$ 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 $\alpha$ : Find all subgroups $H$ 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 $\alpha$ : Find all subgroups $H$ 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 $\alpha$ : Find all subgroups $H$ 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:

potentially operator: Given a 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: Given a group property $\alpha$ and a 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: Given a 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: Given a group property $\alpha$ and 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

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	yes	yes	yes	yes	yes	no
group of prime power order	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: