Normal-extensible automorphisms problem

From Groupprops
Jump to: navigation, search

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:

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

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]