Charles Megibben

On Direct Sums of Countable Groups and Generalizations

In 1933 Ulm [30] published his paper vhich classified countable primary groups. The original proof of Ulm’s -theorem vas rather complicated, but Zippin [32] immediately improved not only the proof but the theorem itself.

Axiom 3 modules

By introducing the concept of a knice submodule, a refinement of the notion of nice subgroup, we are able to formulate a version of the third axiom of countability appropriate to the study of plocal mixed groups in the spirit of the well-known characterization of totally projective pgroups. Our Axiom 3 modules, in fact, form a class of Zp-modules, encompassing the totally projectives in the torsion case, for which we prove a uniqueness theorem and establish closure under direct summands. Indeed Axiom 3 modules turn out to be precisely the previously classified Warfield modules. But with the added power of the third axiom of countability characterization, we derive numerous new results, including the resolution of a long-standing problem of Warfield and theorems in the vein of familiar criteria due to Kulikov and Pontryagin.

Torsion free groups

In this paper we introduce the class of torsion free k-groups and the notion of a knice subgroup. Torsion free k-groups form a class of groups more extensive than the separable groups of Baer, but they enjoy many of the same closure properties. We establish a role for knice subgroups of torsion free groups analogous to that played by nice subgroups in the study of torsion groups. For example, among the torsion free groups, the balanced projectives are characterized by the fact that they satisfy the third axiom of countability with respect to knice subgroups. Separable groups are characterized as those torsion free k-groups with the property that all finite rank, pure knice subgroups are direct summands. The introduction of these new classes of groups and subgroups is based on a preliminary study of the interplay between primitive elements and *-valuated coproducts. As a by-product of our investigation, new proofs are obtained for many classical results on separable groups. Our techniques lead naturally to the discovery that a balanced subgroup of a completely decomposable group is itself completely decomposable provided the corresponding quotient is a separable group of cardinality not exceeding 81; that is, separable groups of cardinality 81 have balanced projective dimension

A nontransitive, fully transitive primary group

AU groups in this note are additively written p-primary abelian groups. If G is such a group, WC define the subgroup p?G, for all ordinals cy, in the usual inductive manner. If G is reduced and x is a nonzero clement of G, then we define h,(x) to be the first ordinal a such that x E p”G and x

On the Congruence of Subgroups of Totally Projectives

pULIG. We set h,(O) == c/3 and follow the convention CL < co for all ordinals a. \Yith each element s E G we associate its Uht sequence u,(x) = (a,, , aI ,..., a,& ,...) where LY ,, : h,(p%). Um sequences are ordered in the obvious pointwise manner. Following Kaplansky [5], we call a reduced primary group G transitire if for each pair of elements X, y E G such that U,(X) = UC;(y) there exists an automorphism of G mapping x to y and fzdly transitive if UG(x) < V,(y) implies the existence of an endomorphism of G mapping x to y. In [J] it is shown that transitivity implies full transitivity at least when p f 2. The existence of reduced primary groups that are neither transitive nor fully transitive was first established in [6]. -4s indicated by our title, we shall here construct a reduced primary group that is fully transitive but not transitive. We rely heavily on Corner’s construction in [1] of primary groups with certain prescribed endomorphism rings. Unfortunately, Corner’s methods, which are based on ideas originally due to Crawlcy [2], treat only primary groups without elements of infinite height, that is, groups G for which pwG = 0. Hut, of course, groups without elements of infinite height are both transitive and fully transitive. Rather than generalizing Corner’s results to a wider class of groups, we shall use a make-shift construction that yields a group G with p”G cyclic and with appropriately restricted endomorphism ring. An endomorphism C#J of G will be called a small edomorphism if for each positive integer k there exists a nonnegative integer n such that (p”G)[p”] is contained in the kernel of 4. We denote the endomorphism ring of G as End G and the ideal of small endomorphisms

Pure subgroups of torsion-free groups

Let H and K be subgroups of the abelian p-group G. We say that H and K are congruent over G provided there is an automorphism of G mapping H onto K.

Balanced projectives and Axiom 3

In this paper, we show that certain new notions of purity stronger than the classical concept are relevant to the study of torsion-free abelian groups. In particular, implications of *-purity, a concept introduced in one of our recent papers, are investigated. We settle an open question (posed by Nongxa) by proving that the union of an ascending countable sequence of *-pure subgroups is completely decomposable provided the subgroups are. This result is false for ordinary purity. The principal result of the paper, however, deals with E-purity, a concept stronger than *-purity but weaker than the usual notion of strong purity. Our main theorem, which has a number of corollaries including the recent result of Nongxa that strongly pure subgroups of separable groups are again separable, states that a E-pure subgroup of a k-group is itself a k-group. Among other results is the negative resolution of the conjecture (valid in the countable case) that a strongly pure subgroup of a completely decomposable group is again completely decomposable.

On the structure of p-local abelian groups

Abstract A global definition of K-nice subgroup is formulated, which generalizes both the notion of a nice subgroup in the context of torsion groups and the notion of a K-nice submodule used in a recent study of p-local balanced projective groups. It is shown, analogous to Hills treatment of totally projective p-groups, that the (global) balanced projective groups of Warfield are precisely those mixed abelian groups that satisfy the Third Axiom of Countability for K-nice subgroups. A variety of equivalent characterizations of balanced projectives are derived by methods independent of the uniqueness and existence theorems previously established for these groups.

Primary Abelian Groups Whose Countable Subgroups Have Countable Closure

A uniqueness theorem in terms of numerical invariants is established for certain mixed p-local groups, called B-modules, which include both the balanced projective groups and the torsion A-groups. In general, a B-module does not contain a decomposition basis, and when it does, it reduces to a direct sum of an A-group and a balanced projective group. A special class of B-modules, the members of which are called S-modules, constitutes a natural generalization of S-groups and the following fact is proved: If H is an isotype subgroup of the p-local balanced projective group G with GH countable, then H is an S-module.

The sequentially pure projective dimension of global groups with decomposition bases

Two subgroups H and K of G are said to be equivalent if there is an automorphism of G that maps H onto K. Using Martin’s Axiom, we establish an equivalence theorem for countable subgroups of a class of groups called c.c. groups. More precisely, we find necessary and sufficient conditions for two countable subgroups of a c.c. group to be equivalent. A p-primary abelian group G is a c.c. group if every countable subgroup has countable closure in the p-adic topology. The equivalence theorem is then used to obtain some structural results and homological properties of c.c. groups.