Sh:1106

• Paolini, G., & Shelah, S. (2018). The automorphism group of Hall’s universal group. Proc. Amer. Math. Soc., 146(4), 1439–1445. arXiv: 1703.10540 DOI: 10.1090/proc/13836 MR: 3754331
• Abstract:
  We study the automorphism group of Hall’s universal locally finite group $H$. We show that in $Aut(H)$ every subgroup of index $< 2^\omega$ 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 [312] 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 [3] and give a partial answer to Question VI.5 of Kegel and Wehrfritz from [6].
• published version (7p)

@article{Sh:1106,
 author = {Paolini, Gianluca and Shelah, Saharon},
 title = {{The automorphism group of Hall's universal group}},
 journal = {Proc. Amer. Math. Soc.},
 fjournal = {Proceedings of the American Mathematical Society},
 volume = {146},
 number = {4},
 year = {2018},
 pages = {1439--1445},
 issn = {0002-9939},
 mrnumber = {3754331},
 mrclass = {20B27 (03C60 20F50)},
 doi = {10.1090/proc/13836},
 note = {\href{https://arxiv.org/abs/1703.10540}{arXiv: 1703.10540}},
 arxiv_number = {1703.10540}
}