### The automorphism tower of a centerless group without choice

by Kaplan and Shelah. [KpSh:882]

Archive for Math Logic, 2009

For a centerless group G, we can define its automorphism
tower. We define G^{alpha} : G^0=G, G^{alpha +1}=
Aut(G^alpha) and for limit ordinals G^delta =
bigcup_{alpha < delta}G^alpha . Let tau_G be the ordinal when
the sequence stabilizes. Thomas' celebrated theorem says tau_G<
2^{|G|})^{+} and more. If we consider Thomas' proof too set
theoretical, we have here a shorter proof with little set
theory. However, set theoretically we get a parallel theorem without
the axiom of choice. We attach to every element in G^alpha, the
alpha-th member of the automorphism tower of G, a unique
quantifier free type over G (whish is a set of words from G*
< x>). This situation is generalized by defining
``(G,A) is a special pair''.

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