Related papers: Some Results on Polish Groups

In first order logic, it is known that you can define a topology so that the countable models of some theory $T$ form a Polish Space (i.e. completely metrizable second countable space). In this paper we use the Baldwin- Boney Relational…

Let H be a product of countably infinite number of copies of an uncountable Polish space X. Let $\Sigma_\xi$ $(\bar {\Sigma}_\xi)$ be the class of Borel sets of additive class \xi for the product of copies of the discrete topology on X (the…

We investigate whether the group algebra of a finite group over a localisation of the integers is semiperfect. The main result is a necessary and sufficient arithmetic criterion in the ordinary case. In the modular case, we propose a…

In [2], an exhaustive construction is achieved for the class of all 4-dimensional unital division algebras over finite fields of odd order, whose left nucleus is not minimal and whose automorphism group contains Klein's four-group. We…

We compute the Whitehead groups of the associative rings in a class which includes (twisted) formal power series rings and the augmentation localizations of group rings and polynomial rings. For any associative ring A, we obtain an…

In [9] we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper we show that our construction from [9] satisfies several additional properties of interest. We deduce from this that countable…

We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group $\Z_n$. Moreover, we prove…

We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $\Gamma$, there are uncountably many maximal subgroups of…

A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups $L_1$ and $L_2$ with $L_1$ not semiprimitive, we construct an infinite…

The concept of an omnigenous locally finite group was introduced in [2] as a generalization of Hall's universal countable locally finite group. In this paper we show that the class of all countable omnigenous locally finite groups is Borel…

(withdrawn.) For every lambda we give an explicit construction of an Abelian group with no non-trivial automorphisms. In particular the group absolutely has no non-trivial automorphisms, hence is absolutely indecomposable. Earlier we knew a…

In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this…

We deal with the problem of existence of uncountable co-Hopfian abelian groups and (absolute) Hopfian abelian groups. Firstly, we prove that there are no co-Hopfian reduced abelian groups $G$ of size $< \mathfrak{p}$ with infinite…

In this paper, we investigate Polish semigroup topologies on the endomorphism monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$. We introduce a new structural condition, property $\mathbb{XX}$, which…

We prove that every $2$-local automorphism on a finite-dimensional semi-simple Lie algebra $\mathcal{L}$ over an algebraically closed field of characteristic zero is an automorphism. We also show that each finite-dimensional nilpotent Lie…

In recent years, much work has been done to measure and compare the complexity of orbit equivalence relations, especially for certain classes of Polish groups. We start by introducing some language to organize this previous work, namely the…

The set of increasing functions on the rational numbers, equipped with the composition operation, naturally forms a topological semigroup with respect to the topology of pointwise convergence in which a sequence of increasing functions…

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all…

We introduce some canonical topologies induced by actions of topological groups on groups and rings. For $H$ being a group [or a ring] and $G$ a topological group acting on $H$ as automorphisms, we describe the finest group [ring] topology…

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…