Wielandt's automorphism tower theorem

Statement
Suppose $$G$$ is a fact about::centerless group. Consider the automorphism tower of $$G$$, defined as follows:


 * $$G_0 = G$$.
 * $$G_{i+1}$$ is the fact about::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