We show that the automorphism group of Philip Hall's universal locally finite group has ample generics,that is, it admits comeager diagonal conjugacy classes in all dimensions.Consequently, it has the small index property, is not the union of a countable chain of non-open subgroups, and has the automatic continuity property. Also, we discuss some algebraic and topological properties of the automorphism group of Hall universal group. For example, we show that every generic automorphism of Hall universal group is conjugate to all of its powers, and hence has roots of all orders.

中文翻译：

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Hall 的泛群有充足的泛型自同构

我们证明了Philip Hall的泛局部有限群的自同构群具有丰富的泛型，即它在所有维度上都承认comager对角共轭类。因此，它具有小索引性质，不是非可数链的并集。开放子组，并具有自动连续性属性。此外，我们还讨论了霍尔泛群的自同构群的一些代数和拓扑性质。例如，我们证明霍尔泛群的每个泛型自同构都与它的所有幂共轭，因此具有所有阶的根。