<![CDATA[WUJ]]><![CDATA[ejournals@wuj.pl]]>https://www.ejournals.eu/20220525ejournals.eu.article.1734303010610.4467/20842589RM.20.003.1243520<![CDATA[Polish groups, automorphism groups, locally finite groups]]>00ED031001Number10102<![CDATA[Reports on Mathematical Logic]]>03<![CDATA[2020]]>01<![CDATA[Number 55]]>1A01<![CDATA[Gianluca Paolini]]>2A01<![CDATA[Saharon Shelah]]>0101<![CDATA[Some Results on Polish Groups]]>01en001003020PLN03<![CDATA[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.
AMS Subject Classification: 03E15, 20K30, 20B27.]]>012020082015000401https://doi.org/10.4467/20842589RM.20.003.12435