相关论文: Borel Whitehead groups

A proof of the Borel completeness of torsion free abelian groups is presented. This proof differs considerably from the approach of Paolini-Shelah.

Given an aleph_1-free abelian group G we characterize the class C_G of all torsion abelian groups T satisfying Ext(G,T)=0 assuming the continuum hypothesis CH. Moreover, in Godel's constructable universe we prove that this characterizes C_G…

We prove that groups for which every countable subgroup is free ($\aleph_1$-free groups) are n-slender, cm-slender, and lcH-slender. In particular every homomorphism from a completely metrizable group to an $\aleph_1$-free group has an open…

We show that it is consistent that there is a strongly aleph_1-free aleph_1-coseparable group of cardinality aleph_1 which is not aleph_1-separable.

An abelian group is said to be aleph_1-free if all its countable subgroups are free. Our main result is: If R is a ring with R^+ free and |R|<lambda <= 2^{aleph_0}, then there exists an aleph_1-free abelian group G of cardinality lambda…

We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of aleph_omega-free abelian…

The connections between Whitehead groups and uniformization properties were investigated by the third author in [Sh:98]. In particular it was essentially shown there that there is a non-free Whitehead (respectively, aleph_1-coseparable)…

An uncountable $\aleph_1$-free group cannot admit a Polish group topology but an uncountable $\aleph_1$-free abelian group can, as witnessed, for example, by the Baer-Specker group $\mathbb{Z}^\omega$; more strongly, $\mathbb{Z}^\omega$ is…

We prove that the Borel space of torsion-free Abelian groups with domain $\omega$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing…

We like to build Abelian groups (or R-modules) which on the one hand are quite free, say $\aleph_{\omega + 1}$-free, and on the other hand, are complicated in suitable sense. We choose as our test problem having no non-trivial homomorphism…

$\aleph_1$-free groups, abelian groups for which every countable subgroup is free, exhibit a number of interesting algebraic and set-theoretic properties. In this paper, we give a complete proof that the property of being $\aleph_1$-free is…

A word $w$ in a free group is {\em achiral} if for every group $G,$ $G_w=G_{w^{-1}},$ where $G_w$ is the image of the word map $w$ on $G.$ We will give few classes of examples of achiral words. Cocke and Ho asked whether Engel words are…

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…

Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis…

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…

We present a class of abelian groups that exhibit a high degree of freeness while possessing no non-trivial homomorphisms to a canonical free object. Unlike prior investigations, which primarily focused on torsion-free groups, our work…

We study (relative) K-Mittag-Leffler modules, with emphasis on the class K of absolutely pure modules. A final goal is to describe the K-Mittag-Leffler abelian groups as those that are, modulo their torsion part, aleph_1-free, Cor.6.12.…

We show that any non abelian free group $\F$ is strongly $\aleph_0$-homogeneous, i.e. that finite tuples of elements which satisfy the same first-order properties are in the same orbit under $\Aut(\F)$. We give a characterization of…

We construct a non-free but aleph_1-separable, torsion-free abelian group G with a pure free subgroup B such that all subgroups of G disjoint from B are free and such that G/B is divisible. This answers a question of Irwin and shows that a…

We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality aleph_1. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove…