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The automorphism group of Hall's universal group


Authors: Gianluca Paolini and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 146 (2018), 1439-1445
MSC (2010): Primary 20B27, 20F50
DOI: https://doi.org/10.1090/proc/13836
Published electronically: November 7, 2017
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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 $ < 2^{\aleph _0}$ 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 \cong G' \leq H$ such that every $ f \in Aut(G')$ extends to an $ \hat {f} \in Aut(H)$ in such a way that $ f \mapsto \hat {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.


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Additional Information

Gianluca Paolini
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel

Saharon Shelah
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel—and—Department of Mathematics, The State University of New Jersey, Hill Center-Busch Campus, Rutgers, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019

DOI: https://doi.org/10.1090/proc/13836
Received by editor(s): March 30, 2017
Received by editor(s) in revised form: May 22, 2017
Published electronically: November 7, 2017
Additional Notes: This research was partially supported by European Research Council grant 338821. No. 1106 on the second author’s publication list.
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2017 American Mathematical Society

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