  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 into the next using inner automorphisms and one takes a direct
limit at limit stages. The question is whether the process ever terminates in a
fixed point, a group which is isomorphic to its automorphism group by the
natural map. In this talk I will prove that every group has a terminating
automorphism tower. After this, I will discuss the set-theoretic aspects of the
height of the automorphism tower of a group, and sketch my recent proof with
Simon Thomas that it is consistent to have a group whose automorphism tower is
wildly modified by forcing; indeed, in various models of set theory the
automorphism tower of this very same group can be almost arbitrarily specified.
