On definability of completely decomposable torsion-free Abelian groups by certain groups of homomorphisms

Let C be an Abelian group. An Abelian group A from a class X of Abelian groups is said to be _C H-definable in X if, for any group B ∈ X, the isomorphism Hom(C,A) ≅ Hom(C,B) implies that A ≅ B. If every group from X is _C H-definable in X, then X is called an _C H-class. In this paper, we study conditions under which a class of completely decomposable torsion-free Abelian groups is an _C H-class, where C is a vector group.     

On definability of a periodic EndE+-group by its endomorphism group

Let A be a class of Abelian groups, A ∈ A, and End(A) be the additive endomorphism group of the group A. The group A is said to be defined by its endomorphism group in the class {ie208-01} if for every group B ∈ B such that End(B) ≅ End(A) the isomorphism B ≅ A holds. The paper considers the problem of definability of a periodic Abelian group A such that End-End(A) ≅ End(A). The classes of periodical Abelian groups, of divisible Abelian groups, of reduced Abelian groups, of nonreduced Abelian groups, and of all Abelian groups are investigated in this paper. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 123–131, 2007.     

Abelian groups with normal endomorphism rings

A ring is said to be normal if all of its idempotents are central. It is proved that a mixed group A with a normal endomorphism ring contains a pure fully invariant subgroup G ⊕ B, the endomorphism ring of a group G is commutative, and a subgroup B is not always distinguished by a direct summand in A. We describe separable, coperiodic, and other groups with normal endomorphism rings. Also we consider Abelian groups in which the square of the Lie bracket of any two endomorphisms is the zero endomorphism. It is proved that every central invariant subgroup of a group is fully invariant iff the endomorphism ring of the group is commutative.     

Nice bases for primary Abelian groups

Abstract Suppose that A is an Abelian p-group. It is proved that if p^ωA is bounded, then A has a bounded nice basis and if p^ωA is a direct sum of cyclic groups, then A has a nice basis. In particular, all Abelian p-groups of length < ω.2 along with all simply presented Abelian p-groups are equipped with bounded nice bases. It is also shown that if length(A)≤ ω.2 and A/p^ωA is countable, then A possesses a bounded nice basis as well as if length(A)≤ ω.2 and p^ωA is countable, then A possesses a nice basis. Moreover, contrasting with these claims, we demonstrate that if length(A)=ω.2 and A/p^ωA is torsion-complete with finite Ulm-Kaplansky invariants, then A does not have a bounded nice basis. If in addition p^ωA is torsion-complete, then A does not have a nice basis, respectively. Finally, we construct a summable -projective group (thus a summable group with a nice basis) which is not a direct sum of countable groups. This answers in negative our question posed in (Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia, 2005). Keywords: Bounded nice basis, Nice basis, Bounded groups, Direct sums of cyclic groups, Summable groups, -projective groups, Simply presented groups, Σ-groups, Torsion-complete groups, Large subgroups, Countable extensions, Bounded extensions Mathematics Subject Classification: 20K10, 20K15 An erratum to this article is available at .     

Torsion Abelian afi-Groups

This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups.     

Sums of lengtht in Abelian groups

LetG be an Abelian group written additively,B a finite subset ofG, and lett be a positive integer. Fort≦|B|, letB _t denote the set of sums oft distinct elements overB. Furthermore, letK be a subgroup ofG and let σ denote the canonical homomorphism σ:G→G/K. WriteB _t (modB _t) forB _tσ and writeB _t (modK) forBσ. The following addition theorem in groups is proved. LetG be an Abelian group with no 2-torsion and letB a be finite subset ofG. Ift is a positive integer such thatt<|B| then |B _t (modK)|≧|B (modK)| for any finite subgroupK ofG.     

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.     

Characterization of Abelian groups with a minimal generating set

Abstract

We characterize Abelian groups with a minimal generating set: Let τ A denote the maximal torsion subgroup of A. An infinitely generated Abelian group A of cardinality κ has a minimal generating set iff at least one of the following conditions is satisfied:

1. dim(A/pA) = dim(A/qA) = κ for at least two different primes p, q.

2. dim(t A/pt A) = κ for some prime number p.

3. Σ{dim(A/(pA + B)) ∣ dim(A/(pA + B)) < κ} = κ for every finitely generated subgroup B of A.

Moreover, if the group A is uncountable, property (3) can be simplified to (3') Σ{dim(A/pA) ∣ dim(A/pA) < κ} = κ, and if the cardinality of the group A has uncountable cofinality, then A has a minimal generating set iff any of properties (1) and (2) is satisfied.     

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.     

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.     

On the classification of complex tori arising from real Abelian surfaces

Let A′ be an Abelian surface over ℝ and denote by A its complexification. We define an intrinsic volume vol(A) of A and show that there are seven possibilities with respect to the rank of End(A) and if vol(A) is rational or not. We prove that each possibility determines the Picard number and the endomorphism algebra of A′ and A respectively.     

On ascending and subnormal subgroups of infinite factorized groups

We consider an almost hyper-Abellan group G of a finite Abelian sectional rank that is the product of two subgroups A and B. We prove that every subgroup H that belongs to the intersection A ∩ B and is ascending both in A and B is also an ascending subgroup in the group G. We also show that, in the general case, this statement is not true. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 6, pp. 842–848, June, 1997.     

Properly 3-realizable groups

A finitely presented group G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π₁ (K) ≅ G whose universal cover has the proper homotopy type of a 3-manifold (with boundary). We discuss the behavior of this property with respect to amalgamated products, HNN-extensions, and direct products, as well as the independence with respect to the chosen 2-polyhedron. We also exhibit certain classes of groups satisfying this property: finitely generated Abelian groups, (classical) hyperbolic groups, and one-relator groups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 95–103, 2005.     

Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms

Let C be an Abelian group. An Abelian group A in some class of Abelian groups is said to be _C H-definable in the class if, for any group B\in , it follows from the existence of an isomorphism Hom(C,A) Hom(C,B) that there is an isomorphism A B. If every group in is _C H-definable in , then the class is called an _C H-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a _C H-class, where C is a completely decomposable torsion-free Abelian group.     

Abelian groups of monotonic permutations

We study m-transitive representations of Abelian m-groups. Representations are found which mimic a variety A {\mathcal A} of all Abelian m-groups and a variety J {\mathcal J} of m-groups defined by an identity x _*  = x ⁻¹.     

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 A ₁, A ₂ of rank 1 and strongly indecomposable group B of rank 2 such that quasi-homomorphism groups Q ? Hom(A _i , B) and Q ? Hom(B, A _i ) 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.     

Maximal orders of Abelian subgroups in finite simple groups

We bring out upper bounds for the orders of Abelian subgroups in finite simple groups. (For alternating and classical groups, these estimates are, or are nearly, exact.) Precisely, the following result, Theorem A, is proved. Let G be a non-Abelian finite simple group and G L₂ (q), where q=p^t for some prime number p. Suppose A is an Abelian subgroup of G. Then |A|³<|G|. Our proof is based on a classification of finite simple groups. As a consequence we obtain Theorem B, which states that a non-Abelian finite simple group G can be represented as ABA, where A and B are its Abelian subgroups, iff G≌ L₂(2^t) for some t ≥ 2; moreover, |A|-2^t+1, |B|=2^t, and A is cyclic and B an elementary 2-group. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 131–160, March–April, 1999.     

On a Problem Related to Homomorphism Groups in the Theory of Abelian Groups

In this paper, for any reduced Abelian group A whose torsion-free rank is infinite, we construct a countable set A(A) of Abelian groups connected with the group A in a definite way and such that for any two different groups B and C from the set A(A) the groups B and C are isomorphic but Hom(B,X) ? Hom(C,X) for any Abelian group X. The construction of such a set of Abelian groups is closely connected with Problem 34 from L. Fuchs’ book “Infinite Abelian Groups,” Vol. 1.     

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.