# Abelian Group Theory Papers

## Andreas R. Blass

The papers are listed in reverse chronological order

Basic Subgroups and Freeness, a Counterexample, joint with Saharon Shelah (to appear in a memorial volume in honor of Tony Corner, "Contributions to Module Theory" (ed. R. Goebel and B. Goldsmith))

We construct a non-free but aleph_1-separable, torsion-free abelian group G with a pure free subgroup B such that all subgroups of G disjoint from B are free and such that G/B is divisible. This answers a question of Irwin and shows that a theorem of Blass and Irwin cannot be strengthened so as to give an exact analog for torsion-free groups of a result proved for p-groups by Benabdallah and Irwin.

Disjoint, Non-Free Subgroups of Abelian Groups, joint with Saharon Shelah (Set Theory: Recent Trends and Applications (ed. A. Andretta) Quaderni de Matematica 17 (2007) 1-24)

Let G be an abelian group and let k be the smallest rank of any group whose direct sum with a free group is isomorphic to G. If k is uncountable, then G has k pairwise disjoint, non-free subgroups. There is an example where k is countably infinite and G does not have even two disjoint, non-free subgroups.

Ultrafilters and Partial Products of Infinite Cyclic Groups, joint with Saharon Shelah (to appear in Comm. Alg.)

We consider, for infinite cardinals k and m, the additive group of sequences of integers, of length k, with non-zero entries in fewer than m positions. Our main result characterizes, in terms of the cardinals, when one such group can be embedded in another. The proof involves some set-theoretic results, one about families of finite sets and one about families of ultrafilters.

Special Families of Sets and Baer-Specker Groups, joint with John Irwin (Comm. Alg., to appear)

We prove that the Baer-Specker group P (the product of countably many copies of the additive group Z of integers) contains a pure subgroup isomorphic to the direct sum of continuum many copies of itself. We produce (2 to the continuum) non-isomorphic subgroups of P, each isomorphic to its dual group. Finally, we show that the isomorphism type of a generalized product of Z's, the set of functions from I to Z with support of size at most the infinite cardinal k, uniquely determines both the cardinality of I and k (as long as there are no measurable cardinals below or equal to k). All three of these results are obtained using set-theoretic existence theorems, namely the existence of large independent families, large almost disjoint families, and Delta-systems.

Is There a Core Class for Almost Free Groups of Size Aleph_1?, joint with John Irwin (Comm. Alg. 32 (2004) 1189-1200)

We discuss the prospects for finding a "core class," i.e., a well-behaved class of non-free abelian groups of cardinality aleph_1 such that every non-free abelian group of cardinality aleph_1 has a subgroup in the core class.

Specker's Theorem for Noebeling's Group (Proc. Amer. Math. Soc. 130 (2002) 1581-1587)

Specker proved that the group Z^{aleph_0} of integer-valued sequences is far from free; all its homomorphisms to Z factor through finite subproducts. Noebeling proved that the subgroup B consisting of the bounded sequences is free and therefore has many homomorphisms to Z. We prove that all "reasonable" homomorphisms from B to Z factor through finite subproducts. Among the reasonable homomorphisms are all those that are Borel with respect to a natural topology on B. In the absence of the axiom of choice, it is consistent that all homomorphisms are reasonable and therefore that Specker's theorem applies to B as well as to Z^{aleph_0}.

Free Subgroups of the Baer-Specker Group, joint with John Irwin (Comm. Alg. 29 (2001) 5769-5794)

The Baer-Specker group P is the product of countably many copies of the additive group of integers. We are concerned with subgroups of P that are free abelian groups. Among the issues we consider are testing freeness of a subgroup by means of its intersections with other specified subgroups, the relationship between freeness and other "smallness" properties, and constraints on the location of free subgroups within P.

Maximal Pure Independent Sets, joint with John Irwin (Abelian Groups, Rings and Modules (AGRAM 2000 Conference, July, 2000, Perth, Western Australia) (ed. A. V. Kelarev, R. Goebel, K. M. Rangaswamy, P. Schultz, and C. Vinsonhaler) Contemporary Math. 273 (2001) 95-106)

We study, in the context of torsion-free abelian groups G, the sets that are maximal with respect to the property of freely generating a pure subgroup of G. We generalize many but not all of the familiar properties of basic subgroups to the subgroups generated by these "maximal pure independent" sets. For example, we show that any two infinite maximal pure independent sets in the same group have the same cardinality, but we also show that this result can fail badly if "infinite" is omitted.

Basic subgroups and a freeness criterion for torsion-free abelian groups, joint with John Irwin (Abelian Groups and Modules, Interrnational Conference in Dublin, August, 1998)(ed. P. Eklof and R. Goebel), Birkhaeuser Verlag (1999) 247-255

We prove that if an abelian group has a basic subgroup of infinite rank and if every subgroup disjoint from any basic subgroup is free then the group itself is free. We prove several corollaries and related results, including some that do not require the existence of a basic subgroup. Finally, we give an example showing that the hypothesis of infinite rank is needed.

Purity and Reid's Theorem, joint with John Irwin (Abelian Groups and Modules, Interrnational Conference in Dublin, August, 1998)(ed. P. Eklof and R. Goebel), Birkhaeuser Verlag (1999) 241-245

We give conditions under which an abelian group is the sum of two free subgroups, one or both of which are pure.

Baer Meets Baire: Applications of Category Arguments and Descriptive Set Theory to Z^{aleph_0}, joint with John Irwin (Abelian Groups and Modules (ed. K. M. Rangaswamy and D. Arnold) Marcel Dekker 193-202)

We apply the Baire category theorem and other classical results of descriptive set theory to the study of the structure of the group Z^{aleph_0} of infinite sequences of integers and some of its subgroups.

On the Group of Eventually Divisible Integer Sequences, joint with John Irwin (Abelian Groups and Modules (ed. K. M. Rangaswamy and D. Arnold) Marcel Dekker 181-192

Let D be the group, under componentwise addition, of infinite sequences x of integers such that each positive integer divides all but finitely many terms of x. We study properties of D and related groups. We show, among other things, that all infinite-rank summands of D are isomorphic to D and that D has essentially indecomposable subgroups of all coranks from 1 to 2^{aleph_0}. Several of our theorems and proofs involve the notion of a basic subgroup, i.e., a pure, free subgroup with divisible quotient.

Subgroups of the Baer-Specker Group with Few Endomorphisms but Large Dual, joint with Ruediger Goebel (Fund. Math. 149 (1996) 19-29)

Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group Z^{aleph_0} with the following properties. Every endomorphism of G differs by an endomorphism of finite rank from some scalar multiplication. Yet G has uncountably many homomorphisms to Z.

Some Abelian Groups with Free Duals, joint with John Irwin and Greg Schlitt (Abelian Groups and Modules, Proceedings of the Padua Conference, June, 1994 (ed. A. Facchini and C. Menini) (Mathematics and Its Applications 343) Kluwer (1995) 57-66)

Let P be the direct product and S the direct sum of countably many copies of the additive group Z of integers. Specker proved that the dual Hom(P,Z) of P is isomorphic to S. We extend this result to groups G that are "near" P. If G is a pure subgroup of P of index smaller than 2^{\aleph_0}, then the dual of G is isomorphic to S. If P is a subgroup of countable index in a group H and if H has no direct summand isomorphic to S, then the dual of H is free (of finite or countable rank). We also obtain similar results for certain other subgroups of P.

Cardinal Characteristics and the Product of Countably Many Infinite Cyclic Groups (J. Alg. 169 (1994) 512-540)

We study, from a set-theoretic point of view, those subgroups of the infinite direct product Z^{aleph_0} for which all homomorphisms to Z annihilate all but finitely many of the standard unit vectors. Specifically, we relate the smallest possible size of such a subgroup to several of the standard cardinal characteristics of the continuum. We also study some related properties and cardinals, both group-theoretic and set-theoretic. One of the set-theoretic properties and the associated cardinal are combinatorially natural, independently of any connection with algebra.

On the divisible parts of quotient groups (Abelian Group Theory and Related Topics (ed. R. Goebel, P. Hill, and W. Liebert) Contemporary Math. 171 (1994) 37-50)

Techniques of combinatorial set theory are applied to the following algebraic problem. Suppose G is an abelian group such that, for all countable subgroups C, the divisible part of the quotient G/C is countable. What can one conclude about the size of the divisible part of G/K when the cardinality of the subgroup K is a given uncountable cardinal?