# Potentially characteristic not implies normal-potentially 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., potentially characteristic subgroup) need not satisfy the second subgroup property (i.e., normal-potentially characteristic subgroup)
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## Statement

It is possible to have a potentially characteristic subgroup that is not a normal-potentially characteristic subgroup.

## Proof

The proof follows 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, $H$ cannot be characteristic in $K$.
• $H$ is potentially characteristic in $G$: for instance, we can realize $G$ as the $2$-Sylow subgroup of the symmetric group of degree four, in such a way that $H$ becomes characteristic.