Torsion-free Abelian Research Articles

Indecomposable p-Local Torsion-Free Groups with Quadratic and Cubic Splitting Fields

Indecomposable torsion-free p-local Abelian groups of finite rank with quadratic and cubic splitting field K are characterized. As a consequence, for groups with quadratic splitting field K it is proved that K-decomposable p-local torsion-free Abelian groups of finite rank are isomorphic if and only if their endomorphism rings are isomorphic.

On the Quasi-Endomorphism Rings of Quasi-Decomposable Torsion-Free Abelian Groups of Rank 4 with a Strongly Indecomposable Quasi-Summand of Rank 2

We obtain a description of quasi-endomorphism rings of torsion-free Abelian groups G of rank 4 that are quasi-decomposable into a direct sum of groups A1 and A2 of rank 1 and a strongly indecomposable group B of rank 2 in the case where the quasi-homomorphism group ℚ ⨂ Hom(B,A2) has rank 2.

Normal Determinability of Torsion-Free Abelian Groups by Their Holomorphs

We investigate torsion-free Abelian groups that are decomposable into direct sums or direct products of homogeneous groups normally defined by their holomorphs. Properties of normal Abelian subgroups of holomorphs of torsion-free Abelian groups are also studied.

On Determinability of a Completely Decomposable Torsion-Free Abelian Group by its Automorphism Group

In this paper, we consider the question of determinability of an Abelian group by its automorphism group in the class of completely decomposable torsion-free Abelian groups whose direct summands of rank 1 have idempotent types.

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.

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 Some Results of a Torsion-Free Abelian Trace Group

On Some Results of a Torsion-Free Abelian Trace Group

On Some Results of a Torsion-Free Abelian Kernel Group

On Some Results of a Torsion-Free Abelian Kernel Group

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.

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.

Partially Divisible Completions of Rigid Metabelian Pro-p-groups

Previously, the author defined the concept of a rigid (abstract) group. By analogy, a metabelian pro-p-group G is said to be rigid if it contains a normal series of the form G = G1 ≥ G2 ≥ G3 = 1 such that the factor group A = G/G2 is torsion-free Abelian, and G2 being a ZpA-module is torsion-free. An abstract rigid group can be completed and made divisible. Here we do something similar for finitely generated rigid metabelian pro-p-groups. In so doing, we need to exit the class of pro-p-groups, since even the completion of a torsion-free nontrivial Abelian pro-p-group is not a pro-p-group. In order to not complicate the situation, we do not complete a first factor, i.e., the group A. Indeed, A is simply structured: it is isomorphic to a direct sum of copies of Zp. A second factor, i.e., the group G2, is completed to a vector space over a field of fractions of a ring ZpA, in which case the field and the space are endowed with suitable topologies. The main result is giving a description of coordinate groups of irreducible algebraic sets over such a partially divisible topological group.

Algebraic Sets in a Finitely Generated 2-Step Solvable Rigid Pro-p-Group

A 2-step solvable pro-p-group G is said to be rigid if it contains a normal series of the form G = G 1 > G 2 > G 3 = 1 such that the factor group A = G/G 2 is torsionfree Abelian, and the subgroup G 2 is also Abelian and is torsion-free as a ℤ p A-module, where ℤ p A is the group algebra of the group A over the ring of p-adic integers. For instance, free metabelian pro-p-groups of rank ≥ 2 are rigid. We give a description of algebraic sets in an arbitrary finitely generated 2-step solvable rigid pro-p-group G, i.e., sets defined by systems of equations in one variable with coefficients in G.

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.

Quasi-endomorphism rings of almost completely decomposable torsion-free Abelian groups of rank 4 that do not coincide with their pseudo-socles

We obtain a description of quasi-endomorphism rings of almost completely decomposable torsion-free Abelian groups of rank 4 that do not coincide with their pseudo-socles.

A condition of indecomposability for Butler $$\mathrm B (n)$$ B ( n ) -groups

We give a manageable sufficient condition for indecomposability of Butler \(\mathrm B (n)\)-groups, allowing the easy construction of a big family of indecomposable torsionfree Abelian groups of finite rank.

Rigid Metabelian Pro-p-Groups

A metabelian pro-p-group G is rigid if it has a normal series of the form G = G1 ≥ G2 ≥ G3 = 1 such that the factor group A = G/G2 is torsion-free Abelian and C = G2 is torsion-free as a ZpA-module. If G is a non-Abelian group, then the subgroup G2, as well as the given series, is uniquely defined by the properties mentioned. An Abelian pro-p-group is rigid if it is torsion-free, and as G2 we can take either the trivial subgroup or the entire group. We prove that all rigid 2-step solvable pro-p-groups are mutually universally equivalent. Rigid metabelian pro-p-groups can be treated as 2-graded groups with possible gradings (1, 1), (1, 0), and (0, 1). If a group is 2-step solvable, then its grading is (1, 1). For an Abelian group, there are two options: namely, grading (1, 0), if G2 = 1, and grading (0, 1) if G2 = G. A morphism between 2-graded rigid pro-p-groups is a homomorphism $$ \varphi $$ : G → H such that Gi $$ \varphi $$ ≤ Hi. It is shown that in the category of 2-graded rigid pro-p-groups, a coproduct operation exists, and we establish its properties.

Absolute Closedness of Torsion-Free Abelian Groups in the Class of Metabelian 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, every inclusion H ≤ G implies that the dominion of H in G (in M) coincides with H. We deal with dominions in torsion-free Abelian subgroups of metabelian groups. It is proved that every nontrivial torsion-free Abelian group is not absolutely closed in the class of metabelian groups. It is stated that if a torsion-free subgroup H of a metabelian group G and the commutator subgroup G′ have trivial intersection, then the dominion of H in G (in the class of metabelian groups) coincides with H.

Quasi-Endomorphism Rings of Almost Completely Decomposable Torsion-Free Abelian Groups of Rank 4 With Zero Jacobson Radical

In this paper, a description of rings of quasi-endomorphisms of almost completely decomposable torsion-free Abelian groups of rank 4 with zero Jacobson radical is obtained.

Determinability of a Completely Decomposable Block-Rigid Torsion-Free Abelian Group by Its Automorphism Group

In this paper, we consider the question of determinability of an Abelian group by its automorphism group in the class of completely decomposable block-rigid torsion-free Abelian groups.