# Strongly image-potentially characteristic subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup of a group is termed a **strongly image-potentially characteristic subgroup** if there exists a surjective homomorphism such that both the kernel of and are characteristic subgroups of .

## Relation with other properties

### Stronger properties

### 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.