Normal-extensible not implies normal

This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., normal-extensible automorphism) need not satisfy the second automorphism property (i.e., normal automorphism)
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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-extensible automorphism-invariant subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about normal subgroup|Get more facts about normal-extensible automorphism-invariant subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not normal-extensible automorphism-invariant subgroup|View examples of subgroups satisfying property normal subgroup and normal-extensible automorphism-invariant subgroup

Statement

In terms of automorphism properties

A normal-extensible automorphism of a group (i.e., an automorphism that can always be extended for any embedding of the group as a normal subgroup of a bigger group) need not be a normal automorphism, i.e., it need not send every normal subgroup to itself.

In terms of subgroup properties

A normal subgroup of a group need not be a normal-extensible automorphism-invariant subgroup: i.e., there may be normal-extensible automorphisms of the group that do not leave the normal subgroup invariant.

Statement with symbols

We can have a group $G$ and a normal-extensible automorphism $\sigma$ of $G$ that is not a normal automorphism: in other words, there exists a normal subgroup $N$ of $G$ such that $\sigma(N) \ne N$.

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.