Normal-extensible automorphisms problem
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 , , satisfies the conditions of being centerless and maximal in its automorphism group. Thus, every automorphism of is normal-extensible.
- Suppose is a simple non-abelian complete group. Then, 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.
- Normal not implies normal-potentially characteristic: We can have a normal subgroup of a group such that there is no group containing as a normal subgroup and as a characteristic subgroup.
- Normal not implies normal-potentially relatively characteristic