# Automorphism tower of a centerless group

From Groupprops

## Definition

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.

## Facts

- 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.