# 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 $\sigma'$ 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:

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:

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:

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: