W. Wickless

Self-small mixed abelian groups G with G/T(G) finite rank divisible

We investigate the class G of self-small mixed abelian groups G with G/T(G) finite rank divisible and the category QG with objects groups in G and maps quasi-homomorphisms. We employ several existing dualities and equivalences between subcategories of QG and subcategories of the category of locally-free torsion-free finite rank abelian groups and quasi-homomorphisms as well as a new duality in QG and new category equivalences from subcategories of QG to certain rational algebras. To aid in our investigations, we also introduce a new invariant and several notions of type for mixed groups. Our results mostly involve classification of subclasses of G in terms of more well-known objects. In particular, we obtain more or less satisfactory descriptions for rank two groups in G, groups in G analogous to cohesive torsion-free groups and groups in G analogous to Murley groups, underlining the resemblance between the mixed groups in G and locally free torsion-free groups of finite rank. For example, we show that ev...

CATEGORIES OF MIXED AND TORSION-FREE FINITE RANK ABELIAN GROUPS

In this paper “group” always means “abelian group”. For a group G let T = T(G) be the torsion part and, for a prime p, let T p = T p (G), be the p-torsion part of G.

Dualities for torsion-free abelian groups of finite rank

Two well-known dualities have been very useful in the study of torsionfree abelian groups of finite rank: Warlield duality for locally free groups and Arnold duality for quotient divisible groups [Wa, Ar]. In this note we establish a duality, on classes of torsion-free abelian groups of finite rank, which generalizes both Warlield and Arnold duality. Our results were inspired by a recent paper of Fomin [FOG], who constructed a duality which is also a special case of the one presented here. The basic idea is to write a torsion-free abelian group of finite rank (hereafter, “group”) G as a sum G = G, + G,, where G, is locally free and G, is quotient divisible. A dual for G is then obtained by adding the War-field dual of G,, W(G,), and the Arnold dual of Gz, A(G,) (inside of Hom(G, Q)). The idea of breaking up G into a locally free and a quotient divisible part is not a new one-it was investigated by Murley in [Mu]. As noted by Murley, one easy way to obtain G, and Gz is to take a full free subgroup F of G and write G/F= D 0 R, where D is a divisible and R is a reduced torsion group. Then choose subgroups G, and G2 of G containing F so that G,IF = R and G,/F = D. In general, the G, obtained in this way is decidedly non-unique, even up to quasi-isomorphism. It is therefore somewhat surprising that we can form G* = W(G,) + A(G,) to obtain a group which, up to quasi-equality, is independent of the choice ofG,. Further complexity can be introduced due to the fact that the Warlield and Arnold duals treat free and divisible localizations differently, prompting our definition of PX-groups (Definition 2.1). On the category of PX-groups and quasi-homomorphisms (in which objects are quasi-equality classes), G + G* defines an exact contravariant functor (Theorem 2.13). The PX-group G* has a special property, which was also studied by Murley. On the PX-groups with this special property, the “dualizable PX-groups,” * determines a duality (Theorem 3.3), which generalizes the dualities of Warlield, Arnold, and Fomin (Proposition 3.5). We remark 474 OO21-8693/90

SELF-SMALL ABELIAN GROUPS

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Direct Sums of Quotient Divisible Groups

This paper investigates self-small abelian groups of finite torsion free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums

GENERALIZED ENDOPRIMAL ABELIAN GROUPS

Abstract An abelian group G is quotient divisible (qd) if it is of finite torsion-free rank and there exists a free subgroup F ⊂ G with G/F a divisible torsion group. We study the category ∑𝒟 with objects arbitrary direct sums of qd groups and maps quasi-homomorphisms. First we identify the qd-indecomposable groups, that is the indecomposables in the category ∑𝒟. These turn out to be the qd groups G such that, if G ⊃ A ⊕ B ⊃ tG for t > 0, then A or B is finite. The major tool for studying ∑𝒟 is the notion of an admissable quasi-decomposition G ≈ ⊕ G k of a group G ∈ ∑𝒢 into a direct sum of qd-indecomposable subgroups. (See Definition 6 below.) For any qd-indecomposable group A and any admissable quasi-decomposition G ≈ ⊕ G k , the number of G k quasi-isomorphic to A is an invariant of G, denoted σ A (G). The cardinal numbers σ A (G), together with the finite Ulm invariants (G), form a complete set of isomorphism invariants for groups G = ⊕G i ∈ ∑𝒢 with p-rank[G i /T(G i )] ≤ 1 for all p, i. Let G ∈ ∑𝒟 with G = A ⊕ B. If G ≈ ⊕ G k , then there is an admissable quasi-decomposition A ≈ ⊕ A j such that each A j is quasi-isomorphic to some G k(j). We consider the problem: When is an abelian group A with an admissable quasi-decomposition A ≈ ⊕ A j into qd-indecomposable subgroups a direct summand of a group in ∑𝒟? We get a positive answer if A is torsion-free or if the mixed groups A j are all in a class 𝒢 ⊂ 𝒟.

A Class of B(2)-Groups

An n-ary endofunction on an abelian group G is a function f : Gn → G such that f(θg1,…,θgn) = θ f(g1,…,gn) for all endomorphisms θ of G. A group G is endoprimal if, for each natural number n, each n-ary endofunction has the following simple form: for some collection of integers {li : 1 ≤ i ≤ n}. The notion of endoprimality arises from universal algebra in a natural way and has been applied to the study of abelian groups in papers Davey and Pitkethly (97), Kaarli and Marki (99) and Gobel, Kaarli, Marki, and Wallutis (to appear). These papers make the case that the notion of endoprimality can give rise to interesting and tractable classes of abelian groups. We continue working along these lines, adapting our definition to make it more suitable for working with general classes of abelian groups. We study generalized endoprimal (ge) abelian groups. Here every n-ary endofunction is required to be of the form for some collection of central endomorphisms {λi : 1 ≤ i ≤ n} of G. (Note that such a sum is always an endofunction.) We characterize generalized endoprimal abelian groups in a number of cases, in particular for torsion groups, torsion-free finite rank groups G such that E(G) has zero nil radical, and self-small mixed groups of finite torsion-free rank.

A-Solvability and Mixed Abelian Groups

Abstract We consider a subclass of Butler groups, the OB (2) -groups. The class B (2) consists of the groups G which appear in an exact sequence 0 → K → D → G → 0 where D is a finite rank completely decomposable group and K is a rank two pure subgroup of D . The groups G belonging to OB (2)⊆ B (2) satisfy an additional condition called “overlap”. We discuss three questions: (1) When is G strongly indecomposable? (2) To what extent is D determined by G ? (3) Is there a reasonable set of numerical invariants for such a G ? We find definitive answers for these questions in the case of strongly indecomposable balanced OB (2)-groups. We also obtain partial results without the balanced hypothesis.

Homological Properties of Quotient Divisible Abelian Groups

An abelian group A is an S-group (S +-group) if every subgroup B ≤ A of finite index is A-generated (A-solvable). This article discusses some of the differences between torsion-free S-groups and mixed S-groups, and studies (mixed) S- and S +-groups, which are self-small and have finite torsion-free rank.

Purity and Self-Small Groups

Abstract An abelian group A is quotient divisible if its torsion subgroup tA is reduced, and it contains a finitely generated free subgroup F such that A/F is the direct sum of a finite and a divisible torsion group. This paper focuses on homological properties of quotient divisible groups. A group A such that tA is reduced is quotient divisible if and only if it is small with respect to the class of quotient divisible groups. Further results investigate when an A-generated torsion group is A-solvable. The last section discusses quotient divisible groups A such that ℚ ⊗ℤ E(A)/tE(A) is a quasi-Frobenius ring.