Automorphism tower of a centerless group

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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_\alphas 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