Torsion-free Abelian Groups Research Articles

Completely decomposable direct summands of torsion-free abelian groups of finite rank

Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism.

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

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.

Scott sentences for certain groups

We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable $$\varSigma _3$$ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are “computable d- $$\varSigma _2$$ ” (the conjunction of a computable $$\varSigma _2$$ sentence and a computable $$\varPi _2$$ sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank 1. These are exactly the additive subgroups of $$\mathbb {Q}$$ . We show that for some of these groups, the computable $$\varSigma _3$$ Scott sentence is best possible, while for others, there is a computable d- $$\varSigma _2$$ Scott sentence.

On 5–manifolds with free fundamental group and simple boundary links in S5

We classify compact oriented $5$-manifolds with free fundamental group and $\pi_{2}$ a torsion free abelian group in terms of the second homotopy group considered as $\pi_1$-module, the cup product on the second cohomology of the universal covering, and the second Stiefel-Whitney class of the universal covering. We apply this to the classification of simple boundary links of $3$-spheres in $S^5$. Using this we give a complete algebraic picture of closed $5$-manifolds with free fundamental group and trivial second homology group.

Strongly self-absorbing C⁎-dynamical systems, III

In this paper, we accomplish two objectives. Firstly, we extend and improve some results in the theory of (semi-)strongly self-absorbing C⁎-dynamical systems, which was introduced and studied in previous work. In particular, this concerns the theory when restricted to the case where all the semi-strongly self-absorbing actions are assumed to be unitarily regular, which is a mild technical condition. The central result in the first part is a strengthened version of the equivariant McDuff-type theorem, where equivariant tensorial absorption can be achieved with respect to so-called very strong cocycle conjugacy.Secondly, we establish completely new results within the theory. This mainly concerns how equivariantly Z-stable absorption can be reduced to equivariantly UHF-stable absorption with respect to a given semi-strongly self-absorbing action. Combining these abstract results with known uniqueness theorems due to Matui and Izumi–Matui, we obtain the following main result. If G is a torsion-free abelian group and D is one of the known strongly self-absorbing C⁎-algebras, then strongly outer G-actions on D are unique up to (very strong) cocycle conjugacy. This is new even for Z3-actions on the Jiang–Su algebra.

On the definability of completely decomposable torsion-free Abelian groups by endomorphism rings and some groups of homomorphisms

In the present paper,we describe conditions on a vector group C which are necessary and sufficient for the class of completely decomposable torsion-free Abelian groups to be a C EH-class.

On fully inert subgroups of completely decomposable groups

The completely decomposable torsion-free Abelian groups with finitely many homogeneous components for which every fully inert subgroup is commensurable with a fully invariant subgroup are described.

New Degree Spectra of Abelian Groups

We show that for every computable ordinal of the form β=δ+2n+1>1, where δ is zero or a limit ordinal and n∈ω, there exists a torsion-free abelian group having an X-computable copy if and only if X is nonlowβ.

On the Definability of Completely Decomposable Torsion-Free Abelian Groups by Endomorphism Rings and Some Groups of Homomorphisms

В данной работе описаны условия для некоторой векторной группы $C$, необходимые и достаточные для того, чтобы класс вполне разложимых абелевых групп без кручения был $_CEH$-классом. Библиография: 10 названий.

Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups

We consider a representation of quasi-endomorphisms of Abelian torsion-free groups of rank 4 bymatrices of order 4 over the field of rational numbers Q. We obtain a classification for quasi-endomorphism rings of Abelian torsion-free groups of rank 4 quasi-decomposable into a direct sum of groups A1, A2 of rank 1 and strongly indecomposable group B of rank 2 such that quasi-homomorphism groups Q ⊗ Hom(Ai, B) and Q ⊗ Hom(B, Ai) for any i = 1, 2 have rank 1 or are zero. Moreover, for algebras from the classification we present necessary and sufficient conditions for their realization as quasi-endomorphism rings of these groups.

Quasiendomorphism algebras of some quasidecomposable rank 4 torsion-free abelian groups

We classify the quasiendomorphism algebras of rank 4 torsion-free abelian groups quasidecomposable as the direct sum of strongly indecomposable rank 2 groups.

Direct decomposition theory under near-isomorphism for a class of infinite rank torsion-free abelian groups

Abstract Near-isomorphism is known as the right concept for classification theorems in the theory of almost completely decomposable groups. As a natural generalization the authors extended in [6] the notion of near-isomorphism to Abelian groups of arbitrary rank. In this article we investigate non-isomorphic direct decompositions of a class of infinite rank torsion-free abelian groups which were defined in [4] as special epimorphic images of so-called almost rigid groups. A complete classification of such decompositions up to near-isomorphism is given.

A Note on Additive Groups of Some Specific Associative Rings

Abstract Almost complete description of abelian groups (A, +, 0) such that every associative ring R with the additive group A satisfies the condition: every subgroup of A is an ideal of R, is given. Some new results for SR-groups in the case of associative rings are also achieved. The characterization of abelian torsion-free groups of rank one and their direct sums which are not nil-groups is complemented using only elementary methods.

A TORSION-FREE ABELIAN GROUP EXISTS WHOSE QUOTIENT GROUP MODULO THE SQUARE SUBGROUP IS NOT A NIL-GROUP

The first example of a torsion-free abelian group$(A,+,0)$such that the quotient group of$A$modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’,Publ. Math. Debrecen27(1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from$2$to$2^{\aleph _{0}}$is presented. Ring multiplications on$p$-pure subgroups of the additive group of the ring of$p$-adic integers are investigated using only elementary methods.

Classification of a class of torsion-free abelian groups

The class of almost completely decomposable groups with a critical typeset of type (2, 2) and a regulator quotient of exponent ≤ p^2 is shown to have exactly 4 near-isomorphism classes of indecomposable groups. Every group of the class is up to near-isomorphism uniquely a direct sum of these four indecomposable groups.

State-closed groups with bounded conjugacy classes

Let [Formula: see text] be the group of automorphisms of the one-rooted [Formula: see text]-ary tree and [Formula: see text] be a transitive state-closed subgroup of [Formula: see text] with bounded finite conjugacy classes. We prove that the torsion subgroup Tor[Formula: see text] has finite exponent and determine an upper bound for the exponent. In case [Formula: see text] is a prime number, we prove that [Formula: see text] is either a torsion group or a torsion-free abelian group.

一类有限秩Abel群的自同构

本文给出了一类有限秩的具有C2自同构的无挠Abel群。 In this paper, we mainly study a class of finite rank torsion free abelian groups which their automorphism groups are C2.

On the square subgroups of decomposable torsion-free abelian groups of rank three

AbstractThe square subgroup of an abelian group

Dominions in Solvable Groups

The dominion of a subgroup H of a group G in a class M is the set of all elements a ∈ G whose images are equal for all pairs of homomorphisms from G to each group in M that coincide on H. A group H is absolutely closed in a class M if, for any group G in M and any inclusion H ≤ G, the dominion of H in G (with respect to M) coincides with H (i.e., H is closed in G). We prove that every torsion-free nontrivial Abelian group is not absolutely closed in ANc. It is shown that if a subgroup H of G in NcA has trivial intersection with the commutator subgroup G′, then the dominion of H in G (with respect to NcA) coincides with H. It is stated that the study of closed subgroups reduces to treating dominions of finitely generated subgroups of finitely generated groups.

Torsion-free modules with UA-rings of endomorphisms

An associative ring R is called a unique addition ring (UA-ring) if its multiplicative semigroup (R, · ) can be equipped with a unique binary operation+ transforming the triple (R, ·, +) to a ring. An R-module A is said to be an End-UA-module if the endomorphism ring EndR(A) of A is a UA-ring. In the paper, the torsion-free End-UA-modules over commutative Dedekind domains are studied. In some classes of Abelian torsion-free groups, the Abelian groups having UA-endomorphism rings are found.