The automorphism group of Hall’s universal group

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.