The automorphism group of hall’s universal group

Gianluca Paolini, Saharon Shelah

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Abstract

We study the automorphism group of Hall’s universal locally finite group H. We show that in Aut(H) every subgroup of index < 2N0 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 the second author 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 his 1978 work, and give a partial answer to Question VI.5 of Kegel and Wehrfritz from their 1973 work.

Original languageEnglish
Pages (from-to)1439-1445
Number of pages7
JournalProceedings of the American Mathematical Society
Volume146
Issue number4
DOIs
StatePublished - 2018

Bibliographical note

Publisher Copyright:
© 2017 American Mathematical Society.

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