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The automorphism tower of a centerless group without Choice

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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”.

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Correspondence to Itay Kaplan.

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Saharom Shelah would like to thank the United States-Israel Binational Science Foundation for partial support of this research. Publication 882.

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Kaplan, I., Shelah, S. The automorphism tower of a centerless group without Choice. Arch. Math. Logic 48, 799–815 (2009). https://doi.org/10.1007/s00153-009-0154-2

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  • DOI: https://doi.org/10.1007/s00153-009-0154-2

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