# 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.