Normal not implies normal-extensible automorphism-invariant in finite

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., normal-extensible automorphism-invariant subgroup)
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Statement

Statement with symbols

It is possible to have a finite group $G$, a normal subgroup $N$ of $G$, and a normal-extensible automorphism $\sigma$ of $G$ such that $\sigma (N)\neq N$.

Related facts

Weaker facts

• Normal-extensible not implies normal
• Normal-extensible not implies extensible
• Normal-extensible not implies inner

Applications

• Normal not implies semi-strongly potentially relatively characteristic
• Potentially characteristic not implies semi-strongly potentially relatively characteristic
• Normal not implies semi-strongly potentially characteristic, normal not implies strongly potentially characteristic
• Potentially characteristic not implies semi-strongly potentially characteristic, potentially characteristic not implies strongly potentially characteristic

Facts used

1. Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
2. Automorphism group of direct power of simple non-abelian group equals wreath product of automorphism group and symmetric group

Proof

Example of the dihedral group

Further information: dihedral group:D8, subgroup structure of dihedral group:D8

Let $G$ be the dihedral group of order eight. Then, every automorphism of $G$ fixes every element of the center of $G$, and also, the inner automorphism group of $G$ is maximal in the automorphism group of $G$. Thus, by fact (1), every automorphism of $G$ is normal-extensible.

However, there is an automorphism of $G$ that interchanges the two normal Klein four-subgroups. Thus, these two normal subgroups are not invariant under this automorphism, and hence, we have an automorphism of $G$ that is normal-extensible but not normal.

Equivalently, the Klein four-subgroups are examples of normal subgroups that are not normal-extensible automorphism-invariant.

Example involving a simple complete group

Let $S$ be a simple complete group. In other words, $S$ is a centerless simple group such that every automorphism of $S$ is inner. Let $G=S\times S$. By fact (2), the automorphism group of $G$ is the wreath product of $S$ with the symmetric group of degree two, which has $G$, the inner automorphism group, as a subgroup of index two. Moreover, $G$ is centerless. Thus, by fact (1), we get that every automorphism of $G$ is normal-extensible.

However, the coordinate exchange automorphism of $G$, that interchanges the two copies of $S$, is not a normal automorphism because it interchanges these two normal subgroups. Thus, we have an example of a normal-extensible automorphism that is not normal.

Equivalently, either of the direct factors is an example of a normal subgroup that is not normal-extensible automorphism-invariant.