We prove that no quantifier-free formula in the language of group theory can define the aleph(1)-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 lambda is the group of automorphism of a locally finite group of power lambda; secondly, we conjecture that the group of automorphisms of a locally finite group of power lambda has a locally finite subgroup of power lambda, and reduce the problem to a problem on p-groups, thus settling the conjecture in the case lambda = aleph(0).
SOME RESULTS ON POLISH GROUPS
Paolini, G
;
2020-01-01
Abstract
We prove that no quantifier-free formula in the language of group theory can define the aleph(1)-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 lambda is the group of automorphism of a locally finite group of power lambda; secondly, we conjecture that the group of automorphisms of a locally finite group of power lambda has a locally finite subgroup of power lambda, and reduce the problem to a problem on p-groups, thus settling the conjecture in the case lambda = aleph(0).
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