The automorphism tower of a centerless group without Choice

Itay Kaplan*, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


For a centerless group G, we can define its automorphism tower. We define Gα: G0 = G, Gα+1 = Aut(Gα) and for limit ordinals Gδ = ∪α>δ. Let τG be the ordinal when the sequence stabilizes. Thomas' celebrated theorem says τG>(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 τG for all countable groups G (better than the one an analysis of Thomas' proof gives). We attach to every element in Gα, the αth member of the automorphism tower of G, a unique quantifier free type over G (which is a set of words from G *). This situation is generalized by defining "(G, A) is a special pair".

Original languageAmerican English
Pages (from-to)799-815
Number of pages17
JournalArchive for Mathematical Logic
Issue number8
StatePublished - Nov 2009

Bibliographical note

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


  • Automorphism tower
  • Axiom of Choice
  • Centerless group


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