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 $<2^{\omega }$ 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 [Sh:312] it follows that for every countable locally finite $G$ there exists $G\cong G^{\prime }\le H$ such that every $f\in Aut(G^{\prime })$ extends to an $\hat{f}\in Aut(H)$ in such a way that $f\mapsto \hat{f}$ embeds $Aut(G^{\prime })$ 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].

References in corpus (2)

• On the isometry group of the Urysohn space
• Existentially closed locally finite groups