Torsion-Free Groups of Infinite Rank

Fuchs, L.

doi:10.1007/978-3-319-19422-6_13

Cited by 24 publications

(48 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let G be a non-trivial pseudocompact torsion Abelian group. Then G is bounded torsion [7,Corollary 3.9], and so there exists a finite set P of prime numbers such that G = p∈P G p , where G p is a non-trivial (bounded torsion) p-group for every p ∈ P ; see [11,Theorem 17.2].…”

Section: 7 Every Non-trivial Pseudocompact Torsion Abelian Group Ismentioning

confidence: 99%

See 1 more Smart Citation

Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups

Dorantes-Aldama

Shakhmatov

2017

Topology and its Applications

View full text Add to dashboard Cite

Abstract. A space X is selectively sequentially pseudocompact if for every family {Un : n ∈ N} of non-empty open subsets of X, one can choose a point xn ∈ Un for every n ∈ N in such a way that the sequence {xn : n ∈ N} has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topology. Since selectively sequentially pseudocompact spaces are strongly pseudocompact in the sense of García-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of García-Ferreira and Tomita.The symbol N denotes the set of natural numbers and N + = N \ {0} denotes the set of positive integer numbers. The symbol Z denotes the group of integer numbers, R the set of real numbers and T the circle group {e iθ : θ ∈ R} ⊆ R 2 . The symbols ω, ω 1 and c stand for the first infinite cardinal, the first uncountable cardinal and the cardinality of the continuum, respectively. For a set X, the set of all finite subsets of X is denoted by [X] <ω , while [X] ω denotes the set of all countably infinite subsets of X.If X is a subset of a group G, then X is the smallest subgroup of G that contains X. For groups which are not necessary Abelian we use the multiplication notation, while for Abelian groups we always use the additive one. In particular, e denotes the identity element of a group and 0 is used for the zero element of an Abelian group. Recall that an element g of a group G is torsion if g n = e for some positive integer n. A group is torsion if all of its elements are torsion. A group G is bounded torsion if there exists n ∈ N such that g n = e for all g ∈ G.Let G be an Abelian group. The symbol t(G) stands for the subgroup of torsion elements of G. For each n ∈ N, note that nG = {ng : g ∈ G} is a subgroup of G and the map g → ng (g ∈ G) is a homomorphism of G onto nG. For a cardinal τ we denote by G (τ ) the direct sum of τ copies of the group G.We will say that a sequence {x n : n ∈ N} of points in a topological space X:• converges to a point x ∈ X if the set {n ∈ N : x n ∈ W } is finite for every open neighbourhood W of x in X; • is convergent if it converges to some point of X.

show abstract

Section: 7 Every Non-trivial Pseudocompact Torsion Abelian Group Ismentioning

confidence: 99%

“…Since both g and f are monomorphisms, so is h; see [8, Lemma 3.10 (ii)]. Since T σ is divisible, there exists a homomorphism ϕ : G → T σ extending h; see [11,Theorem 21.1]. Since H is essential in G by Lemma 6.1 and h is a monomorphism, ϕ is a monomorphism as well; see [8,Lemma 3.5].…”

Section: Selectively Sequentially Pseudocompact Topologies On Torsionmentioning

confidence: 99%

Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups

Dorantes-Aldama

Shakhmatov

2017

Topology and its Applications

View full text Add to dashboard Cite

show abstract

“…See for example [Fuc70,Rob96]. We also refer the reader to [KTW11,Law10] for some more recent examples relating to divisible groups.…”

Section: Definition 23 (Divisible Groupmentioning

confidence: 99%

Convex analysis in groups and semigroups: a sampler

Borwein

Giladi

2016

Math. Program.

View full text Add to dashboard Cite

show abstract

“…For more details we refer the reader to [5]. By group we will mean an abelian group throughout the paper.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Poor and pi-poor Abelian groups

Alizade

Büyükaşık

2016

Communications in Algebra

View full text Add to dashboard Cite

Abstract. In this paper, poor abelian groups are characterized. It is proved that an abelian group is poor if and only if its torsion part contains a direct summand isomorphic to ⊕ p∈P Zp, where P is the set of prime integers. We also prove that pi-poor abelian groups exist. Namely, it is proved that the direct sum of U (N) , where U ranges over all nonisomorphic uniform abelian groups, is pi-poor. Moreover, for a pi-poor abelian group M , it is shown that M can not be torsion, and each p-primary component of M is unbounded. Finally, we show that there are pi-poor groups which are not poor, and vise versa.

show abstract

Torsion-Free Groups of Infinite Rank

Cited by 24 publications

References 0 publications

Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups

Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups

Convex analysis in groups and semigroups: a sampler

Poor and pi-poor Abelian groups

Contact Info

Product

Resources

About