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., semi-strongly potentially characteristic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about potentially characteristic subgroup|Get more facts about semi-strongly potentially characteristic subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property potentially characteristic subgroup but not semi-strongly potentially characteristic subgroup|View examples of subgroups satisfying property potentially characteristic subgroup and semi-strongly potentially characteristic subgroup

Statement

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

Related facts

Weaker facts

• Normal not implies normal-potentially characteristic
• Normal not implies characteristic-potentially characteristic
• Potentially characteristic not implies characteristic-potentially characteristic

Facts used

1. Normal-extensible not implies normal: Actually, we need a slightly stronger version of this statement. Namely, we need that there is a normal-extensible automorphism of a finite group that is not a normal automorphism. The example given in the proof is a finite group, so the proof actually shows the stronger version.
2. Finite normal implies potentially characteristic

Proof

By fact (1) (or rather, the stronger version) there exists a finite group $K$, a normal-extensible automorphism $\sigma$ of $K$, and a normal subgroup $H$ of $K$ such that $\sigma (H)\ne H$.

• $H$ is potentially characteristic in $K$: This follows directly from fact (2).
• $H$ is not semi-strongly potentially characteristic in $K$: Suppose there exists a group $G$ containing $K$ as a normal subgroup. Then, since $\sigma$ is normal-extensible, $\sigma$ extends to an automorphism ${\sigma }^{\prime }$ of $G$. But ${\sigma }^{\prime }(H)=\sigma (H)\ne H$, so $H$ is not characteristic in $G$. Thus, there is no group containing $K$ as a normal subgroup and $H$ as a characteristic subgroup.