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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