Potentially characteristic implies normal

Property-theoretic statement
The subgroup property of being a potentially characteristic subgroup is stronger than the subgroup property of being a normal subgroup.

Verbal statement
Any potentially characteristic subgroup of a group is also a normal subgroup.

Potentially characteristic subgroup
A subgroup $$H$$ of a group $$K$$ is termed potentially characteristic if there exists a group $$G$$ containing $$K$$, such that $$H$$ is a characteristic subgroup inside $$G$$.

Facts used
We use two facts in the proof:


 * Every characteristic subgroup is normal
 * Normality satisfies intermediate subgroup condition: If a subgroup is normal in the whole group, then it is normal in every intermediate subgroup.

Hands-on proof
Given: A group $$K$$, and a potentially characteristic subgroup $$H$$ of $$K$$

To prove: $$H$$ is a normal subgroup of $$K$$

Proof: By the definition of potentially characteristic, there exists a group $$G$$ containing $$K$$ such that $$H$$ is characteristic inside $$G$$.

Since every characteristic subgroup is normal, $$H$$ is a normal subgroup of $$G$$.

Since $$H \le K \le G$$, and normality satisfies intermediate subgroup condition, $$H$$ is also normal in $$K$$. This completes the proof.