## 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^{δ} = ∪_{α>δ}. Let τ_{G} be the ordinal when the sequence stabilizes. Thomas' celebrated theorem says τ_{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 *). This situation is generalized by defining "(G, A) is a special pair".

Original language | American English |
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Pages (from-to) | 799-815 |

Number of pages | 17 |

Journal | Archive for Mathematical Logic |

Volume | 48 |

Issue number | 8 |

DOIs | |

State | Published - 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.

## Keywords

- Automorphism tower
- Axiom of Choice
- Centerless group