# Normal not implies normal-potentially relatively characteristic

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., normal-potentially relatively characteristic subgroup)
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## Statement

A normal subgroup of a group need not be a normal-potentially relatively characteristic subgroup.

## Proof

The proof follows directly from facts (1) and (2).

### Example of the dihedral group

Further information: dihedral group:D8

Let $G$ be the dihedral group of order eight, and $H$ be one of the Klein four-subgroups.

• $H$ is not a normal-potentially characteristic subgroup of $G$: Using the fact that every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible, every automorphism of $G$ can be extended to an automorphism of $K$ for any group $K$ containing $G$ as a normal subgroup. But since there is an automorphism of $G$ not sending $H$ to itself, $H$ is not invariant under this automorphism, which extends to an automorphism of $K$.
• $H$ is normal in $G$: This is obvious.