The automorphism group of Hall's Universal group

by Paolini and Shelah. [PaSh:1106]

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' <= 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].

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