Patrick Keef

ON m,n-BALANCED PROJECTIVE AND m,n-TOTALLY PROJECTIVE PRIMARY ABELIAN GROUPS

If m and n are non-negative integers, then three new classes of abelian p-groups are defined and studied: the m,n-simply presented groups, the m,n-balanced projective groups and the m,n-totally projec- tive groups. These notions combine and generalize both the theories of simply presented groups and p !+n -projective groups. If m,n = 0, these all agree with the class of totally projective groups, but when m+n � 1, they also include the p !+m+n -projective groups. These classes are related to the (strongly) n-simply presented and (strongly) n-balanced projective groups considered in (15) and the n-summable groups considered in (2). The groups in these classes whose lengths are less than ! 2 are character- ized, and if in addition we have n = 0, they are determined by isometries of their p m -socles.

Some isomorphisms of abelian groups involving the Tor functor

Given a reduced group G, the class of groups A such that A Tor(A, G) is studied. A complete characterization is obtained when G is separable.

On the S-equivalence of some general sets of matrices

To help classify the set of square matrices over a ring R under the relation of S-equi valence there is defined a module Av together with a pairing on its torsion submodule, which is referred to as the Seifert system of V. It is shown that if R is a field, or R is a PID and det (tV — V) has content 1, then the Seifert system characterizes an 5-equi valence class. Furthermore, over a field 5-equivalence is reducible to the notion of congruence.

On products of primary abelian groups

Abstract The t-product of a family {Gi}iϵI of abelian p-groups is the torsion subgroup of ΠiϵIGi, which we denote by ΠiϵItGi. The t-product is, in the homological sense, the direct product in the category of abelian p-groups. Since the usual way of writing a torsion-complete group is as a t-product, the notion of t-product provides a way of generalizing this important class of groups. Various properties of t-products are proven. An important part of the study of direct products is the consideration of their epimorphic images. This is also the case with t-products, where we are able to obtain analogous results. Of particular interest are those epimorphic images which are direct sums of cyclics. Applications are given to the ⊕c-topology. As is frequently the case with homomorphisms defined on products, the index sets will be assumed to be non-measurable.

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.

How Commutative Are Direct Products of Dihedral Groups

Summary If G is a finite group, then Pr(G) is the probability that two randomly selected elements of G commute. So G is abelian iff Pr(G) = 1. For any positive integer m, we show that there is a group G which is a direct product of dihedral groups such that Pr(G) = 1/m. We also show that there is a dihedral group G such that Pr(G) = m/m′, where m′ is relatively prime to m.

Representable Preradicals with Enough Projectives

An exact sequence of abelian groups \( 0 \to Z\xrightarrow{\phi }T \to H \to 0 \) said to represent the preradical S if for each abelian group G, SG is the image of Hom(T, G → Hom (Z, G) ≅ G. The preradical S is said to be p-coprimary if H can be chosen to be a p-group. An exposition of some of the main features of representable preradicals is given. An S-pure sequence is an element of SExt(A, B), and this notion allows us to define S-projectivity and S-injectivity. For each ordinal α, p α is a representable p-coprimary preradical with enough projectives. A solution is given to a classical problem of Nunke which asks if all such preradicals are of this form. The height of S is defined to be the height of φ (1) ∈ T, so that the height of p α is α. It is shown that if S is a representable p-coprimary preradical of height α ≤ ω with enough projectives, then S = p α , but examples are constructed of such preradicals of height ω + 1 which do not equal p ω+1.

Classes of primary abelian groups determined by valuated subgroups

Abstract If j and k are non-negative integers, a large class of abelian p -groups is defined, containing both the p ω + j -projectives and the p ω + k -injectives, which is closed with respect to the usual categorical operations. These groups are determined using isometries of their p j + k +1 -socles.

A theorem on the closure of Ω-pure subgroups of CΩ groups in the Ω-topology

The closure of ascending unions of Ω-pure subgroups of CΩ groups is discussed. Applications are given to the torsion product, the balanced-projective dimension of CΩ groups, and Kurepas Hypothesis.

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.