# The Automorphism Group of Hall's Universal Group

## TL;DR

This paper investigates the automorphism group of Hall's universal locally finite group, proving its completeness, characterizing subgroups, and embedding automorphism groups of certain subgroups, thus resolving several open questions.

## Contribution

It establishes that Aut(H) is complete, characterizes subgroups of Aut(H), and embeds automorphism groups of certain countable locally finite groups into Aut(H).

## Key findings

- Aut(H) is complete.
- Subgroups of Aut(H) of index less than continuum are uniquely associated with finite subgroups.
- Automorphism groups of certain subgroups embed into Aut(H).

## 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 $G \cong G' \leq H$ 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].