On the Question of Definability of Homogeneously Decomposable Torsion-Free Abelian Groups by Their Homomorphism Groups and Endomorphism Rings

Mathematical Notes - Let $$C$$ be an Abelian group. A class $$X$$ of Abelian groups is called a $$_CH $$ -class (a $$_CEH$$ -class) if, for any groups $$A$$ and $$B$$ in the class $$X$$ , the...     

Transfinite Diameter with Respect to the Neumann Function

In this paper, we study holomorphically isomorphic Abelian groups, i.e., Abelian groups with isomorphic holomorphs. We also study a generalization of the concept of holomorphic isomorphism, namely, almost holomorphic isomorphism, which is deeply connected with normal Abelian subgroups of holomorphs of Abelian groups. Torsion-free Abelian groups that are determined by their holomorphs are highlighted from different classes. In particular, it has been found that any homogeneous separable group can be determined by its holomorph in the class of all Abelian groups.     

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.     

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.     

On Spectral Analysis and Spectral Synthesis in the Space of Tempered Functions on Discrete Abelian Groups

We consider some problems of spectral analysis and spectral synthesis in the topological vector space \({{\mathcal {M}}}(G)\) of tempered functions on a discrete Abelian group G. It is proved that spectral analysis holds in the space \({{\mathcal {M}}}(G)\) on every Abelian group G, that is, every nonzero closed linear translation invariant subspace of \({{\mathcal {M}}}(G)\) contains an exponential. For any finitely generated Abelian group G it is proved, that spectral synthesis holds in \({{\mathcal {M}}}(G)\), that is, every closed linear translation invariant subspace \({{\mathscr {H}}}\) of \({{\mathcal {M}}}(G)\) coincides with the closed linear span of all exponential monomials belonging to \({{\mathscr {H}}}\). For any Abelian group G with infinite torsion free rank it is proved that spectral synthesis fails to hold in the space \({{\mathcal {M}}}(G)\).     

The Jacobson Radical of an Endomorphism Ring for an Abelian Group

The Jacobson radical of an endomorphism ring is computed for a completely decomposable torsion-free Abelian group and for a mixed Abelian group in one class of mixed groups. For the latter case, we also look into the question when a factor ring w.r.t. the radical is regular in the sense of Nuemann.     

Determination of the Direct Sums of Rational Groups by H-Representations of the Endomorphism Rings up to Equality

The problem of determination of Abelian groups (up to isomorphism) by their rings of endomorphisms in the class of completely decomposable torsion-free Abelian groups has been solved earlier. For the class of direct sums of rational groups, one can speak of determination of Abelian groups by rational representations of their endomorphism rings up to equality. In this paper, we consider this problem for the class of finite direct sums of rational groups and for some subclasses.     

Definability of Completely Decomposable Torsion-Free Abelian Groups by Endomorphism Semigroups and Homomorphism Groups

Mathematical Notes - Let C be an Abelian group. A class X of Abelian groups is called a CE?H-class if, for every groups A, B ∈ X, the isomorphisms E?(A) ? E?(B) and...     

Splitting length of abelian mixed groups of torsion-free rank 1

The splitting length of a mixed Abelian group G is defined as the smallest positive integer n such that \( \mathop \otimes \limits^n G \) splits. The task of determining the splitting length of mixed Abelian groups was formulated by Irwin, Khabbaz, and Rayna. In this paper, a criterion for determining whether \( \mathop \otimes \limits^n G \) splits for countable mixed Abelian groups G of torsion-free rank 1 is found.     

Isomorphism of a Ring to the Endomorphism Ring of an Abelian Group

This paper presents necessary and sufficient conditions under which isomorphism of endomorphism rings of additive groups of arbitrary associative rings with 1 implies isomorphism of these rings. For a certain class of Abelian groups, we present a criterion which shows when isomorphism of their endomorphism rings implies isomorphism of these groups. We demonstrate necessary and sufficient conditions under which an arbitrary ring is the endomorphism ring of an Abelian group. This solves Problem 84 in L. Fuchs’ “Infinite Abelian Groups.”__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 231–234, 2003.     

Nonreduced Abelian groups with UA-rings of endomorphisms

A ring K is a unique addition ring (a UA-ring) if its multiplicative semigroup (K, · ) can be equipped with a unique binary operation + transforming this semigroup to a ring (K, ·, +). An Abelian group is called an End-UA-group if its endomorphism ring is a UA-ring. In the paper, we find End-UA-groups in the class of nonreduced Abelian groups.     

On the computational complexity of the theory of Abelian groups

We develop a series of Ehrenfeucht games and prove the following results:

• 1.(i) The first order theory of the divisible and indecomposable p-group, the first order theory of the group of rational numbers with denominators prime to p and the first order theory of a cyclic group of prime power order can be decided in 2^{2cn log n} Turing time.
• 2.(ii) The first order theory of the direct sum of countably many infinite cyclic groups, the first order theory of finite Abelian groups and the first order theory of all Abelian groups can be decided in 2^2dn Turing space.

     

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.     

Algebraically closed and existentially closed Abelian lattice-ordered groups

If \({\mathcal{G}}\) is an Abelian lattice-ordered (l-) group, then \({\mathcal{G}}\) is algebraically (existentially) closed just in case every finite system of l-group equations (equations and inequations), involving elements of \({\mathcal{G}}\), that is solvable in some Abelian l-group extending \({\mathcal{G}}\) is solvable already in \({\mathcal{G}}\). This paper establishes two systems of axioms for algebraically (existentially) closed Abelian l-groups, one more convenient for modeltheoretic applications and the other, discovered by Weispfenning, more convenient for algebraic applications. Among the model-theoretic applications are quantifierelimination results for various kinds of existential formulas, a new proof of the amalgamation property for Abelian l-groups, Nullstellensätze in Abelian l-groups, and the display of continuum-many elementary-equivalence classes of existentially closed Archimedean l-groups. The algebraic applications include demonstrations that the class of algebraically closed Abelian l-groups is a torsion class closed under arbitrary products, that the class of l-ideals of existentially closed Abelian l-groups is a radical class closed under binary products, and that various classes of existentially closed Abelian l-groups are closed under bounded Boolean products.     

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

Mathematical Notes - 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...     

Finite Abelian Groups with the Strong Endomorphism Kernel Property

An endomorphism h of a group G is said to be strong whenever for every congruence θ on G, (x, y) ∈ θ implies (h(x), h(y)) ∈ θ for every x, y ∈ G. A group G is said to have the strong endomorphism kernel property if every congruence on G is the kernel of a strong endomorphism. In this note, we study the strong endomorphism kernel property in the class of Abelian groups. In particular, we show that a finite Abelian group has the strong endomorphism kernel property if and only if it is cyclic.     

Groups all of whose infinite abelian pd-subgroups are normal

The author studies groups in which any infinite Abelian pd-subgroup (p is a prime) is normal, on the assumption that the group indeed contains such subgroups (IH_p-groups). Necessary and sufficient conditions are established for a group to be an IH_p-group. Relationships are established between the class of IH_p-groups and the class of groups in which all infinite Abelian subgroups are normal, and the class of groups in which all pd-subgroups are normal.Translated from Ukrainskii Matematicheskii Zhurnal, Vo. 44, No. 6, pp. 796–800, June, 1992.     

有限群的共轭类个数与群的性质

用群的共轭类个数刻画了交换群,同时用一个很简洁的方法重新证明了Frobenius G提出的一个著名问题:对于一个固定的数自然数n,共轭类数为n的有限群,在同构的意义下是有限的.