Potentially characteristic implies normal

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property must also satisfy the second subgroup property
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Statement

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.

Definitions used

Potentially characteristic subgroup

Further information: 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.

Proof

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