Automorphism tower of a centerless group
Let be a centerless group. The automorphism tower of is defined as a transfinite tower where:
- For any successor ordinal , is the automorphism group of and the map is the natural homomorphism from the group to its automorphism group given by the conjugation action.
- For any limit ordinal , is the direct limit of the s for with the specified mappings.
By the fact that the centralizer of the inner automorphism group in the automorphism group is trivial for a centerless group, i.e., is trivial if is centerless, we obtain that the automorphism group of a centerless group is centerless. In particular, all the maps are injective.
We say that the automorphism tower of terminates or stabilizes at the ordinal if the inclusion is an isomorphism. Note that this is equivalent to saying that is a complete group.
- Wielandt's automorphism tower theorem: This states that for a finite centerless group, the automorphism tower terminates in finitely many steps.
- Centerless and characteristic in automorphism group implies automorphism group is complete: This gives a special condition under which the tower terminates quickly, after just one step.