Wielandt's automorphism tower theorem
From Groupprops
Statement
Suppose is a Centerless group (?). Consider the automorphism tower of
, defined as follows:
-
.
-
is the Automorphism group (?) of
, with the homomorphism from
to
being the one arising by the natural conjugation action.
Then, there exists a natural number such that
is a complete group: the map from
to
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 s form an ascending chain of subgroups.