Potentially characteristic subgroups characterization problem

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

Statement

Given a group $K$, the goal of the problem is to characterize, for each of the properties given below, all the subgroups satisfying that property:

• Strongly potentially characteristic subgroup: Find all subgroups $H$ of $K$ for which there exists a group $G$ containing $K$ such that both $H$ and $K$ are characteristic subgroups of $K$.
• Semi-strongly potentially characteristic subgroup: Find all subgroups $H$ of $K$ for which there exists a group $G$ containing $K$ such that $H$ is characteristic in $G$ and $K$ is normal in $G$.
• Potentially characteristic subgroup: Find all subgroups $H$ of $K$ for which there exists a group $G$ containing $K$ such that $H$ is characteristic in $G$.
• Potentially relatively characteristic subgroup: Find all subgroups $H$ of $K$ for which there exists a group $G$ containing $K$ such that every automorphism of $G$ that restricts to an automorphism of $K$ also restricts to an automorphism of $H$.

These properties are listed above in decreasing order of strength, and they all lie between characteristic subgroup and normal subgroup. It is possible that they are all equal to the property of normality; the NPC-conjecture states, for instance, that being potentially characteristic is equal to being normal.

Particular cases

NPC-conjecture

Further information: NPC-conjecture

This conjecture states that every potentially characteristic subgroup of a group is normal. Partial progress on the conjecture includes the following: it is true for finite groups, abelian groups, and nilpotent groups. More generally, finite normal implies potentially characteristic, central implies potentially characteristic, and normal subgroup contained in upper central series member is potentially characteristic. However, all these rely on an amalgam construction, that requires the use of infinite groups even in cases where the original groups are finite.