On locally free Abelian groups

The following result is proved in the paper. An Abelian group A is Lw₁, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 121–126.Original Russian Text Copyright © 2005 by E. G. Sklyarenko.This revised version was published online in April 2005 with a corrected issue number.     

Quasipure injective torsion-free groups with indecomposable pure subgroups

In a quasipure injective torsion-free Abelian group whose pure subgroups are strongly indecomposable, any nonzero endomorphism is shown to be a monomorphism. The results of this paper together with results obtained earlier describe quasipure injective torsion-free groups. As a corollary, it is proved that any quasipure injective torsion-free group is transitive. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 587–592, October, 2000.     

Partitions of groups into absolutely dense subsets

It is proved that any infinite Abelian group with finitely many elements of order two can be partitioned into two subsets that are dense in any nondiscrete group topology, and hence contain no cosets of infinite subgroups. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 706–711, May, 2000.     

Rings of quasi-endomorphisms of strongly indecomposable torsion-free Abelian groups

In the paper the classification of rings of quasi-endomorphisms of strongly indecomposable Abelian torsion-free groups of rank three is obtained. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 763–773, May, 1998.     

Algebraic sets in metabelian groups

The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = F_n is a free metabelian group of rank n; (2) G = W_n,k is a wreath product of free Abelian groups of ranks n and k. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 503–513, July–August, 2007.     

Definability of vector groups by endomorphism semigroups

Some conditions for torsion-free Abelian groups to be isomorphic are found; the groups in question are decomposable into a direct product of rank 1 groups with isomorphic endomorphism semigroups.Translated fromAlgebra i Logika, Vol. 33, No. 4, pp. 422–428, July-August, 1994.     

Divisible rigid groups

A soluble group G is rigid if it contains a normal series of the form G = G₁ > G₂ > … > G_p > G_p+1 = 1, whose quotients G_i/G_i+1 are Abelian and are torsion-free as right ℤ[G/G_i]-modules. The concept of a rigid group appeared in studying algebraic geometry over groups that are close to free soluble. In the class of all rigid groups, we distinguish divisible groups the elements of whose quotients G_i/G_i+1 are divisible by any elements of respective groups rings Z[G/G_i]. It is reasonable to suppose that algebraic geometry over divisible rigid groups is rather well structured. Abstract properties of such groups are investigated. It is proved that in every divisible rigid group H that contains G as a subgroup, there is a minimal divisible subgroup including G, which we call a divisible closure of G in H. Among divisible closures of G are divisible completions of G that are distinguished by some natural condition. It is shown that a divisible completion is defined uniquely up to G-isomorphism. Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1). Translated from Algebra i Logika, Vol. 47, No. 6, pp. 762–776, November–December, 2008.     

Diagonal invariant ideals of Toeplitz algebras on discrete groups

Diagonal invariant ideals of Toeplitz algebras defined on discrete groups are introduced and studied. In terms of isometric representations of Toeplitz algebras associated with quasi-ordered groups, a character of a discrete group to be amenable is clarified. It is proved that whenG is Abelian, a closed twosided non-trivial ideal of the Toeplitz algebra defined on a discrete Abelian ordered group is diagonal invariant if and only if it is invariant in the sense of Adji and Murphy, thus a new proof of their result is given.     

Essential subgroups of topological groups

We study the class Wof Hausdorff topological groups Gfor which the following two cardinal invariants coincide

ES(G)=min{|H|:H≤ Gdense and essential}

TD(G)=min{|H|:H≤ Gtotally dense}

We prove that W contains the following classes:locally compact abelian groups, compact connected groups, countably compact totally discon¬nected abelian groups, topologically simple groups, locally compact Abelian groups when endowed with their Bohr topology, totally minimal abelian groups and free Abelian topological groups. For all these classes we are also able to giv ean explicit computation of the common value of ESand TD.     

Summability of a ring of endomorphisms of vector groups

Sufficient conditions are found for arbitrary endomorphisms to have a presentation by sum of automorphisms in the class of Abelian groups whose endomorphism rings are isomorphic to a ring of finite-rowed matrices. This result is then used to establish a criterion for such a presentation in terms of type language for vector Abelian groups, whose reduced part does not exceed the first cardinal of nonzero measure. Translated fromAlgebra i Logika, Vol. 37, No. 1, pp. 88–100, January–February, 1998.     

Abelian groups

In the present, third survey of reviews of articles on Abelian groups are included papers reviewed during 1972–1978. Here, as in the previous surveys, the questions concerning finite Abelian groups, topological groups, ordered groups, group algebras, modules, structures of subgroups, as well as the questions connected with logic, are not touched on.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 17, pp. 3–63, 1979.     

Definability of separable torsion-free Abelian groups by endomorphism semigroups

In the class of separable torsion-free Abelian groups, we describe those which are defined by their endomorphism semigroups.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 523–530, September-October, 1995.     

Topological groups in which each nowhere dense subset is closed

Assuming the validity of the combinatorial principlep=c, which follows from Martin's axiom, it is proved that an arbitrary nondiscrete metrizable group topology on an Abelian group can be strengthened to a nondiscrete group topology in which each nowhere dense subset is closed.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 207–211, August, 1998.     

Abelian groups

The present, fourth survey of review articles on Abelian groups includes works reviewed in the years 1979–1984. Also, as in the preceding surveys, no attention has been given to questions involving finite Abelian groups, topological groups, ordered groups, group algebras, modules, or the structure of subgroups, or to questions connected with logic. The word group throughout is understood to mean Abelian group (except in the case of the group of automorphisms of an Abelian group). Concepts and notation not defined in this survey can be found in the books [117, 118]. The letterZ throughout denotes the group (or ring) of integers,Q the group (or field) of rational numbers,Q _p the group (ring) of all rational numbers whose denominator is not divisible by the primep, I_p the group of allp-adic integers. The torsion part of an Abelian groupG is throughout denoted bytG. Translated from Itogi Nauki i Tekhniki. Seriya Algebra, Topologiya, Geometriya, Vol. 23, pp. 51–118.     

Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups

An Abelian group A is called correct if for any Abelian group B isomorphisms A ≅ B′ and B ≅ A′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism A ≅ B. We say that a group A is determined by its subgroups (its proper subgroups) if for any group B the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups A and B such that corresponding subgroups are isomorphic implies A ≅ B. In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.     

Characterization of generalized Chernikov groups among groups with involutions

The class of generalized Chernikov groups is characterized, i.e., the class of periodic locally solvable groups with the primary ascending chain condition. The name of the class is related to the fact that the structure of such groups is close to that of Chernikov groups. Namely, a Chernikov group is defined as a finite extension of a direct product of finitely many quasi-cyclic groups, and a generalized Chernikov group is a layer-finite extension of a direct productA of quasi-cyclicp-groups with finitely many factors for each primep such that each of its elements does not commute elementwise with only finitely many Sylow subgroups ofA. A theorem that characterizes the generalized Chernikov groups in the class of groups with involution is proved. Translated fromMatematicheskie Zametki, Vol. 62, No 4, pp. 577–587, October, 1997. Translated by A. I. Shtern     

On pureness in Abelian groups

Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the module _E (G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary ring R with the identity element. Indeed, if _R M is an arbitrary unitary left module and M ⁺ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M ⁺) under the canonical ring homomorphism R → E(M ⁺). Then it holds that if _E(M+) N is a pure submodule in _E(M+) M ⁺, then _R N is a pure submodule in _R M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238, 2004.     

Homomorphic images of Abelian groups

For some classes of Abelian groups, we answer the question of when a union of homomorphic images of a group is a subgroup of another group. In connection with this, the concept of a homomorphically stable group is introduced and properties of homomorphically stable groups from different classes of Abelian groups are studied. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 17–24, 2007.     

Abelian torsion groups with a countably compact group topology

Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (3) (1994) 323–337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm–Kaplansky invariants.We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811–837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1–3) (2005) 2–54] showed this equivalence for groups of cardinality not greater than .We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality κ^ω, for any infinite cardinal κ. In particular, it is consistent that for every cardinal κ there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ>κ.