Primary Abelian Research Articles

Ulm Function Analysis of Full Transitivity in Primary Abelian Groups

This research addresses the problem posed by Chekhlov and Danchev (2015) regarding variations of Kaplansky's full transitivity in primary abelian groups 𝐺. By delving into three distinct forms of full transitivity within the endomorphism ring of 𝐺, specifically focusing on subgroups, subrings, and unitary subrings generated by commutator endomorphisms, we aim to provide a comprehensive understanding of the totally projective groups exhibiting these properties. The Ulm function of 𝐺 emerges as a key tool in solving this problem and related inquiries, leading to a precise characterization of the groups involved.

On height sequences in primary Abelian groups

For an abelian p-group G, the cyclic subgroups X and Y are defined to be quotient-Ulm equivalent if the quotients G/X and G/Y have the same Ulm function. A complete characterization is given for when X and Y are factor-Ulm equivalent in terms of their respective height sequences. For groups that are quotient-transitive or CS-transitive, this is used to unify, generalize and simplify work of Goldsmith, Gong and Strüngmann (Quotient-Transitivity and Cyclic Subgroup-Transitivity, in J. Group Theory, and Cyclic Subgroup Transitivity for Abelian Groups, to appear in Rend. Semin. Mat. Univ. Padova). These results pertain to the class of transitive groups that satisfy the property satisfy the property that G/X and G/Y are isomorphic whenever they have the same Ulm function. This extensive class contains both the separable groups, as well as the totally projective groups.

МАТРИЧНОЕ ПРЕДСТАВЛЕНИЕ ЭНДОМОРФИЗМОВ ПРИМАРНЫХ ГРУПП МАЛЫХ РАНГОВ

For endomorphism rings of finite primary Abelian groups of rank 2 and 3, generalized matrix rings isomorphic to them are constructed. In each of these matrix rings, necessary and sufficient conditions for the invertibility of matrices are found, as well as formulas for constructing the inverse matrix.

Retraction

The Editor-in-Chief was notified that Section 2 of the paper: Alveera Mehdi, Ayazul Hasan and Fahad Sikander, On submodules of QTAG-modules, International Journal of Engineering, Science and Technology, Vol. 6, No. 2 (2014), pp. 20-30. was copied from the paper “Peter V. Danchev, Generalized Dieudonné and Honda Criteria for Primary Abelian Groups, Acta Math. Univ. Comenianae LXXVII, no. 1 (2008), 85-91.”.As the reporter puts it: “It was a very clever kind of plagiarism”… “It is similar with other terminology, which helps them [the authors] cover up the plagiarism and protects them from detection by Google search.”Subsequent to examination of these articles and the supporting evidence, and after discussion with the corresponding author of the article and the Editorial Board member in that field, the claim is correct and the Editor-in-Chief of the International Journal of Engineering, Science and Technology has decided to retract the paper immediately.”

Primary Abelian Groups Isomorphic to the Proper Fully Invariant Subgroup

We study primary Abelian groups containing proper fully invariant subgroup isomorphic to the group. The admissable sequence of the Ulm–Kaplansky invariants for primary groups is used. Using these sequences, IF-groups are described in some important classes of Abelian p-group.

Extensions of nice bases on Ulm subgroups of primary Abelian groups with totally projective quotients

SupposeG is an abelian p-group, α is an ordinal andG/pG is totally projective. We show that G has a nice basis if and only if pG has a nice basis. This extends results of Danchev in Bull. Malaysian Math. Sci. Soc. (2010), Bull. Allah. Math. Soc. (2011) and An. St. Univ. Ovidius Constanta (2012), as well as joint work with Keef in Rocky Mount. J. Math. (2011).

Classes of Subgroups of Simply Presented Primary Abelian Groups

A class 𝒳 of abelian p-groups is closed under ω1-bijective homomorphisms if whenever f: G → H is a homomorphism with countable kernel and cokernel, then G ∈ 𝒳 iff H ∈ 𝒳. For an ordinal α, we consider the smallest class with this property containing (a) the p α-bounded simply presented groups; (b) the p α-projective groups; (c) the subgroups of p α-bounded simply presented groups. This builds upon classical results of Nunke from [14] and [15]. Particular attention is paid to the separable groups in these classes.

An Extension of Nice Bases on Ulm Subgroups of Primary Abelian Groups

Abstract Suppose A is an Abelian p-group and is an ordinal such that A/pα A is a direct sum of countable groups. It is shown that A has a nice basis if, and only if, pα A has a nice basis. This strengthens an earlier result of ours in Bull. Allah. Math. Soc. (2011) proved when α< ω2.

Fully Invariant Subgroups of n-Summable Primary Abelian Groups

Fully Invariant Subgroups of n-Summable Primary Abelian Groups

N-Summable Valuated p n -Socles and Primary Abelian Groups

Generalizing the classical concept of a valuated vector space, we introduce the notion of a valuated p n -socle. A valuated p n -socle is said to be n-summable if it is isometric to the valuated direct sum of countable valuated groups. Many properties of these objects are established, and in particular, they are shown to be completely classifiable using Ulm invariants, providing a strong connection with the theory of direct sums of countable abelian p-groups. The resulting theory is then applied to the category of primary abelian groups.

An Application of Set Theory to ω + n-Totally p ω+n -Projective Primary Abelian Groups

Several equivalent descriptions are given of the class of primary abelian groups whose separable subgroups are all direct sums of cyclic groups; such groups are called ω-totally Σ-cyclic. This establishes the converse of a theorem due to Megibben. For n < ω, this is generalized to a consideration of the class of primary abelian groups whose pω+n-bounded subgroups are all pω+n-projective. The question of whether there are such groups that are proper in the sense that they are neither pω+n-projective nor ω-totally Σ-cyclic is shown to be logically equivalent to a natural question about the structure of valuated vector spaces. Finally, it is shown that both of these statements are independent of ZFC.

On the Jacobson radical of the endomorphism ring of a homogeneous separable group

Pierce [1] posed the problem of describing the elements of the Jacobson radical of the endomorphism ring of a primary Abelian group in terms of actions of these elements on the group. The problem of describing the Jacobson radical of the endomorphism ring of a homogeneous separable group was posed in this sense in [2, Problem 18]. In the present note, we study the Jacobson radical of the endomorphism ring of a homogeneous separable torsion-free group, and the above problem is solved (see Theorem 4 and Theorem 8). All groups under consideration are Abelian torsion-free groups. Let G be a group; denote by E(G) its endomorphism ring. Denote by H(G) the set of all endomorphisms α ∈ E(G) such that αx =0 for any x ∈ D(G) ,w hereD(G) is the divisible part of the group G and, for any x ∈ G \ D(G), the inequality hp(x) h p(x) ,w herehp(x) stands for the p-height of the element x. Denote by F (G) the set of all endomorphisms of G whose image has finite rank. The Jacobson radical of the endomorphism ring of G is denoted by J(E(G)). In what follows, we consider torsion-free Abelian groups only. For the notation and terms which are not defined in this note; see [2] and [3]. As is known [2, Sec. 22], if G is a homogeneous completely decomposable group, then J(E(G)) = H(G) ∩ F (G). For a homogeneous separable group, only the following inclusion relations are known: H(G) ∩ F (G) ⊆ J(E(G)) ⊆ H(G).

On Primary Abelian Groups Modulo Finite Subgroups

We study the existence of several classes 𝒦 of Abelian p-groups, p a fixed prime, which possess the following property: A ∈ 𝒦⇔A/F ∈ 𝒦, whenever F is a finite subgroup of the Abelian p-group A.

Notes on Essentially Finitely Indecomposable Nonthick Primary Abelian Groups

We show that there exists an essentially finitely indecomposable abelian p-group which is not a thick group, thus resolving once again in the negative an Irwin's conjecture which was already settled by Dugas and Irwin (1992) and Danchev (2008). Some other related questions are studied as well.

Nice bases for primary Abelian groups

Suppose that A is an Abelian p-group. It is proved that if p ω A is bounded, then A has a bounded nice basis and if p ω A 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 ω A is countable, then A possesses a bounded nice basis as well as if length(A)≤ ω.2 and p ω A is countable, then A possesses a nice basis. Moreover, contrasting with these claims, we demonstrate that if length(A)=ω.2 and A/p ω A is torsion-complete with finite Ulm-Kaplansky invariants, then A does not have a bounded nice basis. If in addition p ω A is torsion-complete, then A does not have a nice basis, respectively. Finally, we construct a summable $p^{\omega+2}$ -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, $p^{\omega+n}$ -projective groups, Simply presented groups, Σ-groups, Torsion-complete groups, Large subgroups, Countable extensions, Bounded extensions Mathematics Subject Classification: 20K10, 20K15

On the torsion product and direct sums of primary cyclic groups

A classical question of R. Nunke asks when the torsion product of two primary Abelian groups is a direct sum of cyclic groups. For each ordinal α, we define a partially ordered set, P α , on which there is a natural multiplication, and a function μ α from the class of all primary Abelian groups to P α . This construction is used to address Nunke's problem and a complete answer is given, at least for groups whose cardinality is less than the first weakly inaccessible cardinal.

Nice bases for primary Abelian groups

Nice bases for primary Abelian groups

Slenderness, Completions, and Duality for Primary Abelian Groups

IfAis a fixed abelian group with endomorphism ringE, then for any groupG, letG*=Hom(G,A) and for anyE-moduleM, letM*=HomE(M,A). The evaluation map σG:G→G** is defined in the usual way andGisA-reflexive if σGis an isomorphism. This is strongly related to the question of whetherAis slender as anE-module, and we discuss thep-groups for which this holds. In some important cases,G** can be viewed as the completion ofGin a linear topology. It is known that ifA=⊕,nZpn, andGis ap-group of non-measurable cardinality, thenG** can be identified with the completion ofGin the ⊕c-topology, and we provide a generalization of this result. We also show that for any groupNof non-measurable cardinality there is a groupGsuch thatG**/σG(G)≅N.

Primary abelian groups admitting only small homomorphisms

Primary abelian groups admitting only small homomorphisms

On Generalizations of Purity in Primary Abelian Groups

On Generalizations of Purity in Primary Abelian Groups