Strongly image-potentially characteristic subgroup

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Definition

A subgroup $H$ of a group $G$ is termed a strongly image-potentially characteristic subgroup if there exists a surjective homomorphism $\rho:K \to G$ such that both the kernel of $\rho$ and $\rho^{-1}(H)$ are characteristic subgroups of $K$.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Kernel of a characteristic action on an abelian group

Weaker properties

• Semi-strongly image-potentially characteristic subgroup
• Image-potentially characteristic subgroup
• Normal subgroup

Facts

• Finite NIPC theorem: This shows that any normal subgroup of a finite group is strongly image-potentially characteristic.
• No nontrivial abelian normal p-subgroup for some prime p implies every normal subgroup is strongly image-potentially characteristic: This shows that for a very large class of groups, every normal subgroup is strongly image-potentially characteristic.