# Every nontrivial normal subgroup is potentially characteristic-and-not-intermediately characteristic

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 an intermediately characteristic subgroup of (i.e., it is not characteristic in every intermediate subgroup).

## Facts used

- Every nontrivial normal subgroup is potentially normal-and-not-characteristic
- NPC theorem: This statems that every normal subgroup is potentially characteristic.

## Proof

**Given**: A group , a nontrivial normal subgroup of .

**To prove**: There exists a group containing such that is characteristic in but not characteristic in every intermediate subgroup.

**Proof**:

- By fact (1), there exists a group containing such that is normal and not characteristic in .
- By fact (2), there exists a group containing such that is characteristic in .

Thus, is characteristic in but not in the intermediate subgroup , completing the proof.