## The automorphism group of Hall’s universal group

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- by Gianluca Paolini and Saharon Shelah PDF
- Proc. Amer. Math. Soc.
**146**(2018), 1439-1445 Request permission

## 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 $G \cong G’ \leqslant H$ 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.## References

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## Additional Information

**Gianluca Paolini**- Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel
- MR Author ID: 1110693
**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
- MR Author ID: 160185
- ORCID: 0000-0003-0462-3152
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**146**(2018), 1439-1445 - MSC (2010): Primary 20B27, 20F50
- DOI: https://doi.org/10.1090/proc/13836
- MathSciNet review: 3754331