# Wielandt's automorphism tower theorem

## Statement

Suppose is a Centerless group (?). Consider the automorphism tower of , defined as follows:

- .
- is the Automorphism group (?) of , with the homomorphism from to being the one arising by the natural conjugation action.

Then, there exists a natural number such that is a complete group: the map from to is an isomorphism.

Note that for a centerless group, the inner automorphism group is centralizer-free in the automorphism group. Thus, the automorphism group is again centerless. Further, since we know that the natural map from a centerless group to its automorphism group is injective (it identifies the group with its inner automorphisms), we obtain that the s form an ascending chain of subgroups.