The automorphism tower of a centerless group without Choice

Abstract

For a centerless group G, we can define its automorphism tower. We define G α : G 0 = G, G α+1 = Aut(G α ) and for limit ordinals $${G^{\delta}=\bigcup_{\alpha<\delta}G^{\alpha}}$$ . Let τ 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 (using Fodor’s lemma), we have here a more direct proof with little set theory. However, set theoretically we get a parallel theorem without the Axiom of Choice. Moreover, we give a descriptive set theoretic approach for calculating an upper bound for τ G for all countable groups G (better than the one an analysis of Thomas’ proof gives). We attach to every element in G α , the αth member of the automorphism tower of G, a unique quantifier free type over G (which is a set of words from $${G*\langle x\rangle}$$ ). This situation is generalized by defining “(G, A) is a special pair”.