@article{4606e83f-4e73-401f-9fcc-68ceb5f0f16b,
author = {Gianluca  Paolini, Saharon Shelah},
title = {Some Results on Polish Groups},
journal = {Reports on Mathematical Logic},
volume = {2020},
number = {Number 55},
year = {2020},
issn = {0137-2904},
pages = {61-71},keywords = {Polish groups; automorphism groups; locally finite groups},
abstract = {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.},
doi = {10.4467/20842589RM.20.003.12435},
url = {https://ejournals.eu/en/journal/reports-on-mathematical-logic/article/some-results-on-polish-groups}
}