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

## Statement

Suppose $G$ is a group and $H$ is a nontrivial Normal subgroup (?) of $G$. Then, there exists a group $K$ containing $G$ such that $H$ is a Characteristic subgroup (?) of $K$ but not an intermediately characteristic subgroup of $K$ (i.e., it is not characteristic in every intermediate subgroup).

## Facts used

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

## Proof

Given: A group $G$, a nontrivial normal subgroup $H$ of $G$.

To prove: There exists a group $K$ containing $G$ such that $H$ is characteristic in $K$ but not characteristic in every intermediate subgroup.

Proof:

1. By fact (1), there exists a group $L$ containing $G$ such that $H$ is normal and not characteristic in $L$.
2. By fact (2), there exists a group $K$ containing $L$ such that $H$ is characteristic in $K$.

Thus, $H$ is characteristic in $K$ but not in the intermediate subgroup $L$, completing the proof.