# Every nontrivial normal subgroup is potentially characteristic-and-not-fully invariant

From Groupprops

## Statement

Suppose is a group and is a nontrivial Normal subgroup (?) of . Then, there exists a group containing such that is a Characteristic subgroup (?) of but not a Fully invariant subgroup (?) of .

## Related facts

- Every nontrivial characteristic subgroup is potentially characteristic-and-not-fully invariant
- Characteristic not implies fully invariant
- NPC theorem: This states that every normal subgroup can be realized as a characteristic subgroup in some bigger group.
- Normal not implies potentially fully invariant