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., $C_{\mathrm{A}\mathrm{u}\mathrm{t}(G)}\mathrm{I}\mathrm{n}\mathrm{n}(G)$ 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.