Submission date: 30 March 2017

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 in Aut(G') extends to an \hat{f} in Aut(H) in such a way that f → \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].

Mathematics Subject Classification: 20B27, 20F50

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Full text arXiv 1703.10540: pdf, ps.