Automorphism tower of a centerless group

Definition
Let $$G$$ be a centerless group. The automorphism tower of $$G$$ is defined as a transfinite tower $$G_\alpha$$ where:


 * $$G_0 = G$$
 * For any successor ordinal $$\alpha + 1$$, $$G_{\alpha + 1}$$ is the automorphism group of $$G_\alpha$$ and the map $$G_\alpha \to G_{\alpha + 1}$$ is the natural homomorphism from the group to its automorphism group given by the conjugation action.
 * For any limit ordinal $$\gamma$$, $$G_\gamma$$ is the direct limit of the $$G_\alpha$$s for $$\alpha < \gamma$$ 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., $$C_{\operatorname{Aut}(G)}\operatorname{Inn}(G)$$ is trivial if $$G$$ is centerless, we obtain that the automorphism group of a centerless group is centerless. In particular, all the maps $$G_\alpha \to G_{\alpha + 1}$$ are injective.

We say that the automorphism tower of $$G$$ terminates or stabilizes at the ordinal $$\alpha$$ if the inclusion $$G_\alpha \to G_{\alpha + 1}$$ is an isomorphism. Note that this is equivalent to saying that $$G_\alpha$$ 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.