Wielandt's automorphism tower theorem

Statement

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

• $G_0 = G$.
• $G_{i+1}$ is the Automorphism group (?) of $G_i$, with the homomorphism from $G_i$ to $G_{i+1}$ being the one arising by the natural conjugation action.

Then, there exists a natural number $n$ such that $G_n$ is a complete group: the map from $G_n$ to $G_{n+1}$ 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 $G_i$s form an ascending chain of subgroups.

Related facts

• Centerless implies inner automorphism group is centralizer-free in automorphism group
• Centerless and characteristic in automorphism group implies automorphism group is complete
• Characteristically simple and non-abelian implies automorphism group is complete