The automorphism tower of a centerless group (mostly) without choice

2006/06/09 by Itay Kaplan, Saharon Shelah, Kaplan, Itay +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Logic (math.LO) #math.GR #math.LO #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0606216

published as Arch. Math. Logic 48 No. 8 (2009) 799--815

arxiv created 2006/06/09 · openalex publication_date 2006/06/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a centerless group G, we can define its automorphism tower. We define G^alpha : G⁰=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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