Class Of Abelian Groups Research Articles

Filial rings on direct sums and direct products of torsion-free abelian groups

A ring whose additive group coincides with an abelian group G is called a ring on G. An abelian group G is called a TI-group if every associative ring on G is filial. If every (associative) ring on an abelian group G is an SI-ring (a hamiltonian ring), then G is called an SI-group (an SI_H-group). In this article, TI-groups, SI_H-groups and SI-groups are described in the following classes of abelian groups: almost completely decomposable groups, separable torsionfree groups and non-measurable vector groups. Moreover, a complete description of non-reduced TI-groups, SI_H-groups and SI-groups is given. This allows us to only consider reduced groups when studying TI-groups.

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.

Completely Decomposable Quotient Divisible Abelian Groups with Isomorphic Endomorphism Semigroups

Let $$\Lambda$$ be a class of Abelian groups. A group $$A\in\Lambda$$ is said to be determined by its endomorphism semigroup $$E^\star(A)$$ in the class $$\Lambda$$ if every isomorphism $$E^\star(A)\cong E^\star(B)$$ , where $$B\in\Lambda$$ , implies the isomorphism $$A\cong B$$ . The paper describes those Abelian groups in the class $$\mathscr Q\mathscr D_{\mathrm{cd}}$$ of completely decomposable quotient divisible Abelian groups which are determined by their endomorphism semigroups in the class $$\mathscr Q\mathscr D_{\mathrm{cd}}$$ .

On uniformly fully inert subgroups of abelian groups

AbstractIf H is a subgroup of an abelian group G and φ ∈ End(G), H is called φ-inert (and φ is H-inertial) if φ(H) ∩ H has finite index in the image φ(H). The notion of φ-inert subgroup arose and was investigated in a relevant way in the study of the so called intrinsic entropy of an endomorphism φ, while inertial endo-morphisms (these are endomorphisms that are H-inertial for every subgroup H) were intensively studied by Rinauro and the first named author.A subgroup H of an abelian group G is said to be fully inert if it is φ-inert for every φ ∈ End(G). This property, inspired by the “dual” notion of inertial endomorphism, has been deeply investigated for many different types of groups G. It has been proved that in some cases all fully inert subgroups of an abelian group G are commensurable with a fully invariant subgroup of G (e.g., when G is free or a direct sum of cyclic p-groups). One can strengthen the notion of fully inert subgroup by defining H to be uniformly fully inert if there exists a positive integer n such that |(H + φH)/H| ≤ n for every φ ∈ End(G). The aim of this paper is to study the uniformly fully inert subgroups of abelian groups. A natural question arising in this investigation is whether such a subgroup is commensurable with a fully invariant subgroup. This paper provides a positive answer to this question for groups belonging to several classes of abelian groups.

Abelian RE-Groups

An Abelian group on which every nonzero ring is isomorphic to the ring of endomorphisms of this group is called an RE-group. In the present paper, the RE-groups are described in some classes of Abelian groups, including periodic, divisible, unreduced, and torsion-free rank-1 groups. It is shown that there are no RE-groups in the class of completely decomposable torsion-free Abelian groups.

Groups whose non-permutable subgroups are metaquasihamiltonian

Abstract If 𝔛 {\mathfrak{X}} is a class of groups, define a sequence of group classes 𝔛 1 , 𝔛 2 , … , 𝔛 k , … {\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots,\mathfrak{X}_{k},\ldots} by putting 𝔛 1 = 𝔛 {\mathfrak{X}_{1}=\mathfrak{X}} and choosing 𝔛 k + 1 {\mathfrak{X}_{k+1}} as the class of all groups whose non-permutable subgroups belong to 𝔛 k {\mathfrak{X}_{k}} . In particular, if 𝔄 {\mathfrak{A}} is the class of abelian groups, 𝔄 2 {\mathfrak{A}_{2}} is the class of quasimetahamiltonian groups, i.e. groups whose non-permutable subgroups are abelian. The aim of this paper is to study the structure of 𝔛 k {\mathfrak{X}_{k}} -groups, with special emphasis on the case 𝔛 = 𝔄 {\mathfrak{X}=\mathfrak{A}} . Among other results, it will also be proved that a group has a finite normal subgroup with quasihamiltonian quotient if and only if it is locally graded and belongs to 𝔄 k {\mathfrak{A}_{k}} for some positive integer k.

GROUPS WHOSE NONNORMAL SUBGROUPS ARE METAHAMILTONIAN

If $\mathfrak{X}$ is a class of groups, we define a sequence $\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots ,\mathfrak{X}_{k},\ldots$ of group classes by putting $\mathfrak{X}_{1}=\mathfrak{X}$ and choosing $\mathfrak{X}_{k+1}$ as the class of all groups whose nonnormal subgroups belong to $\mathfrak{X}_{k}$. In particular, if $\mathfrak{A}$ is the class of abelian groups, $\mathfrak{A}_{2}$ is the class of metahamiltonian groups, that is, groups whose nonnormal subgroups are abelian. The aim of this paper is to study the structure of $\mathfrak{X}_{k}$-groups, with special emphasis on the case $\mathfrak{X}=\mathfrak{A}$. Among other results, it will be proved that a group has a finite commutator subgroup if and only if it is locally graded and belongs to $\mathfrak{A}_{k}$ for some positive integer $k$.

Algebraic description of limit models in classes of abelian groups

We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation. We show: Theorem 0.1(1)If G is a limit model of cardinality λ in the class of abelian groups with the subgroup relation, thenG≅(⊕λQ)⊕⊕pprime(⊕λZ(p∞)).(2)If G is a limit model of cardinality λ in the class of torsion-free abelian groups with the pure subgroup relation, then:•If the length of the chain has uncountable cofinality, thenG≅(⊕λQ)⊕Πpprime(⊕λZ(p))‾.•If the length of the chain has countable cofinality, then G is not algebraically compact. We also study the class of finitely Butler groups with the pure subgroup relation, we show that it is an AEC, Galois-stable and (<ℵ0)-tame and short.

Canonical and Algebraically Closed Groups in Universal Classes of Abelian Groups

Using sets of finitely generated Abelian groups closed under the discrimination operator, we describe principal universal classes 𝒦 within a quasivariety 𝔄p, the class of groups whose periodic part is a p-group for a prime p. Also the concept of an algebraically closed group in 𝒦 is introduced, and such groups are classified.

Large characteristic subgroups and abstract group classes

It is well-known that every virtually abelian group contains an abelian characteristic subgroup of finite index. We shall say that a group class is F-characteristic if any group containing an -subgroup of finite index has also a characteristic subgroup of finite index that belongs to . Thus the class of abelian groups is F -characteristic. Many other relevant group classes have recently been proved to be F-characteristic, and the aim of this paper is to give a further contribution to this topic, in particular in the context of classes of generalized nilpotent groups.

Abelian Groups With Sufficiently π-Regular Endomorphism Ring

The notion of π-regular endomorphism ring of an abelian group, which generalizes the notion of regular endomorphism ring, was introduced in papers of L. Fuchs and K. Rangaswamy. They described periodic abelian groups with π-regular endomorphism ring and found necessary conditions for an abelian group to have π-regular endomorphism ring. In this paper, we study abelian groups with sufficiently π-regular endomorphism ring, which form a subclass of the class of abelian groups with π-regular endomorphism ring, and find necessary and sufficient conditions for an abelian group to have sufficiently π-regular endomorphism ring.

Subgroups generated by images of endomorphisms of Abelian groups and duality

Abstract A subgroup H of a group G is called endo-generated if it is generated by endo-images, i.e. images of endomorphisms of G. In this paper we determine the following classes of Abelian groups: (a) the endo-groups, i.e. the groups all of whose subgroups are endo-generated; (b) the endo-image simple groups, i.e. the groups such that no proper subgroup is an endo-image; (c) the pure-image simple, i.e. the groups such that no proper pure subgroup is an endo-image; (d) the groups all of whose endo-images are pure subgroups; (e) the ker-gen groups, i.e. the groups all of whose kernels are endo-generated. Some dual notions are also determined.

On Some Classes of Hopfian Abelian Groups and Modules

The hopficity property in a well-known class of Abelian groups—completely decomposable torsion-free groups—as well as in a class of modules over generalized matrix rings is studied. Examples of non-Hopfian completely decomposable torsion-free group are constructed.

Strongly inert subgroups of abelian groups

Mixing in a natural way the notions of fully inert (see \[6]) and strongly invariant (see \[4]) subgroups of abelian groups, we introduce the strongly inert subgroups which we determine for several classes of abelian groups.

The classification of infinite abelian groups with partial decomposition bases in $L_\infty \omega $

We consider the class of abelian groups with partial decomposition bases, which includes groups classified by Ulm, Warfield, Stanton and others. We define an invariant and classify these groups in the language $L_{\infty \omega }$, or equivalently, up to partial isomorphism. This generalizes a result of Barwise and Eklof and builds on Jacoby's classification of local groups with partial decomposition bases in $L_{\infty \omega }$.

Universal Structures

We deal with the existence of universal members in a given cardinality for several classes. First, we deal with classes of abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality or λ=λℵ0. We use versions of being reduced—replacing Q by a subring (defined by a sequence t¯)—and get quite accurate results for the existence of universals in a cardinal, for embeddings and for pure embeddings. Second, we deal with (variants of) the oak property (from a work of Džamonja and the author), a property of complete first-order theories sufficient for the nonexistence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups (naturally interpreted, so for quantifier-free formulas) and deals more with the existence of universals.

Representations of finite posets over the ring of integers modulo a prime power

The classical category Rep$(S,\mathbb {Z}_{p})$ of representations of a finite poset $S$ over the field $\mathbb {Z}_{p}$ is extended to two categories, Rep$(S,\mathbb {Z}_{p^{m}})$ and uRep$(S,\mathbb {Z}_{p^{m}})$, of representations of $S$ over the ring $\mathbb {Z}_{p^{m}}$. A list of values of $S$ and $m$ for which Rep$(S,\mathbb {Z}_{p^{m}})$ or uRep$(S,\mathbb {Z}_{p^{m}})$ has infinite representation type is given for the case that $S$ is a forest. Applications include a computation of the representation type for certain classes of abelian groups, as the category of sincere representations in (uRep$(S,\mathbb {Z}_{p^{m}})$) Rep$(S,Z_{p^{m}})$ has the same representation type as (homocyclic) $(S,p^{m})$-groups, a class of almost completely decomposable groups of finite rank. On the other hand, numerous known lists of examples of indecomposable $(S,p^{m})$-groups give rise to lists of indecomposable representations.

Partial Decomposition Bases and Global Warfield Groups

The class of abelian groups with partial decomposition bases was developed by the first author in order to generalize Barwise and Eklof's classification of torsion groups in L∞ω. In this article, we continue to explore algebraic characteristics of this class and establish a uniqueness theorem, extending our previous work on mixed p-local groups to the global case. It is shown that groups with partial decomposition bases are characterized in terms of Warfield groups and k-groups of Hill and Megibben. In fact, we prove that the class of groups with partial decomposition bases is identical to the class of k-groups, and, as such, closed under direct summands, and that every finitely generated subgroup of a k-group is locally nice. Also, we introduce and explore subgroups possessing a partial subbasis. As an application, it is shown that isotype k-subgroups of abelian groups are k-groups.

Canonical and existential groups in universal classes of Abelian groups

Universal classes of Abelian groups are classified in terms of sets of finitely generated groups closed with respect to the discrimination operator. The notions of a principal universal class and a canonical group for such a class are introduced. For any universal class K, the class K ec of existentially closed groups generated by the universal theory of K is described. It is proved that K ec is axiomatizable and, therefore, the universal theory of K has a model companion.

On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

We introduce two classes of abelian groups which have either only trivial fully invariant subgroups or all their nontrivial (respectively nonzero) fully invariant subgroups are isomorphic, called IFI-groups and strongly IFI-groups, such that every strongly IFI-group is an IFI-group, respectively. Moreover, these classes coincide when the groups are torsion-free, but are different when the groups are torsion as well as, surprisingly, mixed groups cannot be IFI-groups. We also study their important properties as our results somewhat contrast with those from [13] and [14].