Wielandt's automorphism tower theorem

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Statement

Suppose G is a Centerless group (?). Consider the automorphism tower of G, defined as follows:

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_is form an ascending chain of subgroups.

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