  The automorphism tower of a group is obtained by computing its automorphism
group, the automorphism group of THAT group, and so on, iterating
transfinitely. Each group maps canonically into the next using inner
automorphisms, and so at limit stages one can take a direct limit and continue
the iteration. The tower is said to terminate if a fixed point is reached, that
is, if a group is reached which is isomorphic to its automorphism group by the
natural map. This occurs if a complete group is reached, one which is
centerless and has only inner automorphisms. Wielandt [1939] proved the
classical result that the automorphism tower of any centerless finite group
terminates in finitely many steps. Rae and Roseblade [1970] proved that the
automorphism tower of any centerless Cernikov group terminates in finitely many
steps. Hulse [1970] proved that the the automorphism tower of any centerless
polycyclic group terminates in countably many steps. Simon Thomas [1985] proved
that the automorphism tower of any centerless group eventually terminates. In
this paper, I remove the centerless assumption, and prove that every group has
a terminating transfinite automorphism tower.
