Let $G$ be a finite abelian group and $A\subseteq G$. For $n\in G$, denote by $r_{A}(n)$ the number of ordered pairs $(a_{1},a_{2})\in A^{2}$ such that $a_{1}+a_{2}=n$. Among other things, we prove that for any odd number $t\geq 3$, it is not possible to partition $G$ into $t$ disjoint sets $A_{1},A_{2},\dots ,A_{t}$ with $r_{A_{1}}=r_{A_{2}}=\cdots =r_{A_{t}}$.

As a special case of fully invariant subgroups, strongly invariant subgroups are introduced and studied for Abelian groups.

The class of almost completely decomposable groups with a critical typeset of type $(2,2)$ and a homocyclic regulator quotient of exponent $p^{3}$ is shown to be of bounded representation type. There are only $16$ isomorphism at $p$ types of indecomposables, all of rank $8$ or lower.

This paper presents new results concerning the structure of $\text{SI}$-groups and refines and purifies the results obtained in this field by Shalom Feigelstock [‘Additive groups of rings whose subrings are ideals’, Bull. Aust. Math. Soc.55 (1997), 477–481]. The structure theorem describing torsion-free $\text{SI}$-groups is proved in the associative case. Numerous examples of $\text{SI}$-groups are given. Some inconsistencies in Feigelstock’s article are noted and corrected.

We characterize the abelian groups $G$ for which the functors $\mathrm{Ext} (G, - )$ or $\mathrm{Ext} (- , G)$ commute with or invert certain direct sums or direct products.

We prove for a large family of rings R that their λ-pure global dimension is greater than one for each infinite regular cardinal λ. This answers in the negative a problem posed by Rosický. The derived categories of such rings then do not satisfy, for any λ, the Adams λ-representability for morphisms. Equivalently, they are examples of well-generated triangulated categories whose λ-abelianization in the sense of Neeman is not a full functor for any λ. In particular, we show that given a compactly generated triangulated category, one may not be able to find a Rosický functor among the λ-abelianization functors.

In this paper, we characterize quadratic number fields possessing unique factorization in terms of the power cancellation property of torsion-free rank-two abelian groups, in terms of Σ-unique decomposition, in terms of a pair of point set topological properties of Eilenberg–Mac Lane spaces, and in terms of the sequence of rational primes. We give a complete set of topological invariants of abelian groups, we characterize those abelian groups that have the power cancellation property in the category of abelian groups, and we characterize those abelian groups that have Σ-unique decomposition. Our methods can be used to characterize any direct sum decomposition property of an abelian group.

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 are discussed.

The variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

We call an algebraic group monothetic if it possesses a dense cyclic subgroup. For an arbitrary field k we describe the structure of all, not necessarily affine, monothetic k-groups G and determine in which cases G has a k-rational generator.

Valuated Butler groups of finite rank are investigated. The valuated B2-groups are both epic images and pure subgroups of completely decomposable valuated groups of finite rank (Theorem 3.1). However, there are fundamental changes in the theory of Butler groups when valuations are involved. We introduce valuated B1-groups and show that they are valuated B2-groups. Surprisingly, valuated B2-groups of rank greater than 1 need not be valuated B1 -groups, unless they carry a special kind valuation, see Theorem 7.1. Theorem 6.5 gives a full characterization of valuated B1 -groups of finite rank, generalizing Bican's characterization of Butler groups.

In this note we characterize the abelian groups G which have two different proper subgroups N and M such that the subgroup lattice L(G)=[0, M]∪ [N, G] is the union of these intervals.

A group A is said to be endoprimal if its term functions are precisely the functions which permute with all endomorphisms of A. In this paper we describe endoprimal groups in the following three classes of abelian groups: torsion groups, torsionfree groups of rank at most 2, direct sums of a torsion group and a torsionfree group of rank 1.

We solve the following problem which was posed by Barnes in 1962. For which abelian groups G and H of the same prime power order is it possible to embed the subgroup lattice of G in that of H? It follows from Barnes' results and a theorem of Herrmann and Huhn that if there exists such an embedding and G contains three independent elements of order p2, then G and H are isomorphic. This reduces the problem to the case that G is the direct product of cyclic p-groups only two of which have order larger than p. We determine all groups H for which the desired embedding exists.

This note investigates torsion-free abelian groups G of finite rank which embed, as subgroups of finite index, in a finite direct sum C of subgroups of the additive group of rational numbers. Specifically, we examine the relationship between G and C when the index of G in C is minimal. Some properties of Warfield duality are developed and used (in the case that G is locally free) to relate our results to earlier ones by Burkhardt and Lady.

A composition sequence for a torsion-free abelian group A is an increasing sequenceof pure subgroups with rank 1 quotients and union A. Properties of A can be described by the sequence of types of these quotients. For example, if A is uniform, that is all the types in some sequence are equal, then A is complete decomposable if it is homogeneous. If A has finite rank and all permutations ofone of its type sequences can be realized, then A is quasi-isomorphic to a direct sum of uniform groups.

We define an equivalence relation on the class of torsion-free abelian groups under which two groups are equivalent ifevery pure subgroup of one has a non-zero image in the other, and each has a non-zero image in every torsion-free factor of the other.

We study the closure properties of the equivalence classes, and the structural properties of the class of all equivalence classes. Finally we identify a class of groups which satisfy Krull-Schmidt and Jordan-Hölder properties with respect to the equivalence.

Epimorphic images of compact (algebraically compact) abelian groups are called cotorsion groups after Harrison. In a recent paper, Ph. Schultz raised the question whether “cotorsion” is a property which can be recognized by its small cotorsion epimorphic images: If G is a torsion-free group such that every torsion-free reduced homomorphic image of cardinality is cotorsion, is G necessarily cortorsion? In this note we will give some counterexamples to this problem. In fact, there is no cardinal k which is large enough to test cotorsion.

A subgroup H of an abelian p–group G is pure in G if the inclusion map of H into G is an isometry with respect to the (pseudo-) metrics on H and G associated with their p–adic topologies. In this paper, those subgroups (called here imbedded subgroups) of abelian groups for which the inclusion is a homeomorphism with respect to the p–adic topologies are studied, the aim being to compare the concepts of imbeddedness and purity. Perhaps the main results indicate that imbedded subgroups are considerably more abundant than pure subgroups. Groups for which this is not the case are characterized.

A Σ-group is an abelian group on which is given a family of infinite sums having properties suggested by, but weaker than, those which hold for absolutely convergent series of real or complex numbers. Two closely related questions are considered. The first concerns the construction of a Σ-group from an arbitrary abelian group on which certain series are given to be summable, certain of these series being required to sum to zero. This leads to a Σ-theoretic construction of R from Q and in general of the completion of an arbitrary metrizable abelian group (with the associated unconditional sums) from that group. The second question is whether, in a given Σ-group, the values of the infinite sums may be determined solely from a knowledge of which series are summable. Such a Σ-group is said to be relatively free and it is shown that all metrizable abelian groups are relatively free.