Normal-extensible automorphisms problem

Statement

An automorphism $\sigma$ of a group $G$ is termed a normal-extensible automorphism if, whenever $G$ is a normal subgroup of a group $K$, there is an automorphism ${\sigma }^{\prime }$ of $K$ whose restriction to $G$ is $\sigma$.

The normal-extensible automorphisms of any group form a subgroup of the automorphism group. Further, this subgroup contains the group of inner automorphisms, since every inner automorphism is normal-extensible. In fact, it contains the group of extensible automorphisms as well.

The normal-extensible automorphisms problem is the problem of characterizing the group of normal-extensible automorphisms of a group. There are two extreme cases for normal-extensible automorphisms:

• Group in which every automorphism is normal-extensible
• Group in which every normal-extensible automorphism is inner

A group in which every automorphism is inner is at both extremes.

A somewhat intermediate case that is also important is group in which every normal-extensible automorphism is normal.

Groups in which every automorphism is normal-extensible

A group in which every automorphism is inner obviously satisfies the additional condition that every automorphism of the group is normal-extensible. However, there are examples of groups with outer automorphisms in which every automorphism is normal-extensible. Two basic facts:

• Centerless and maximal in automorphism group implies every automorphism is normal-extensible
• Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible

Here are some examples of this:

• In dihedral group:D8, every automorphism is center-fixing and the inner automorphism group is maximal in the automorphism group. Thus, every automorphism is normal-extensible.
• Any alternating group $A_n$, $n\ne 1,2,6$, satisfies the conditions of being centerless and maximal in its automorphism group. Thus, every automorphism of $A_n$ is normal-extensible.
• Suppose $S$ is a simple non-abelian complete group. Then, $S\times S$ is of index two in its automorphism group.

These examples show that:

• Normal-extensible not implies inner
• Normal-extensible not implies normal: The examples of the dihedral group and the square of a simple non-abelian complete group show that there can exist normal-extensible automorphisms that are not normal automorphisms, i.e., they do not send normal subgroups to themselves.

The second fact can be restated as follows: a normal subgroup need not be a normal-extensible automorphism-invariant subgroup. This has the following corollaries:

• Normal not implies normal-potentially characteristic: We can have a normal subgroup $H$ of a group $G$ such that there is no group $K$ containing $G$ as a normal subgroup and $H$ as a characteristic subgroup.
• Normal not implies normal-potentially relatively characteristic

Group in which every normal-extensible automorphism is inner, and normal

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