The Automorphism Group of Hall's Universal Group

2017/03/30 by Gianluca Paolini, Saharon Shelah, Paolini, Gianluca +1 · 1 citation
Mathematics · #20B27 #20F50 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1703.10540

openalex publication_date 2017/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the automorphism group of Hall's universal locally finite group H. We show that in Aut(H) every subgroup of index < 2^ω lies between the pointwise and the setwise stabilizer of a unique finite subgroup A of H, and use this to prove that Aut(H) is complete. We further show that Inn(H) is the largest locally finite normal subgroup of Aut(H). Finally, we observe that from the work of [Sh:312] it follows that for every countable locally finite G there exists G ≅ G' ≤ H such that every f ∈ Aut(G') extends to an f ∈ Aut(H) in such a way that f ↦ f embeds Aut(G') into Aut(H). In particular, we solve the three open questions of Hickin on Aut(H) from [3], and give a partial answer to Question VI.5 of Kegel and Wehrfritz from [6].

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