%0 Journal Article
%T Some Results on Polish Groups
%A Paolini, Gianluca
%A Shelah, Saharon
%J Reports on Mathematical Logic
%V 2020
%R 10.4467/20842589RM.20.003.12435
%N Number 55
%P 61-71
%K Polish groups, automorphism groups, locally finite groups
%@ 0137-2904
%D 2020
%U https://ejournals.eu/en/journal/reports-on-mathematical-logic/article/some-results-on-polish-groups
%X We prove that no quantifier-free formula in the language of group theory can define the ℵ1-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of a given Borel complete class, and observe that this space must contain at least one uncountable group. Finally, we prove some results on the structure of the group of automorphisms of a locally finite group: firstly, we prove that it is not the case that every group of automorphisms of a graph of power λ is the group of automorphism of a locally finite group of power λ; secondly, we conjecture that the group of automorphisms of a locally finite group of power λ has a locally finite subgroup of power λ, and reduce the problem to a problem on p-groups, thus settling the conjecture in the case λ = ℵ0.