Some Results on Polish Groups
Some Results on Polish Groups
Author(s): Gianluca Paolini, Saharon ShelahSubject(s): Logic, Analytic Philosophy
Published by: Wydawnictwo Uniwersytetu Jagiellońskiego
Keywords: Polish groups; automorphism groups; locally finite groups;
Summary/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 fi- nite 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.
Journal: Reports on Mathematical Logic
- Issue Year: 2020
- Issue No: 55
- Page Range: 61-71
- Page Count: 11
- Language: English