On the structure of abelian groups

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1968

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This thesis is divided into three chapters. Each chapter deals with a different aspect of the theory of abelian groups. In the first chapter, techniques are developed for investigating the decomposability of pure subgroups of torsion free groups. It is shown that a torsion free group G is a pure essential extension of a completely bi decomposable subgroup C such that |G| < |C|[raised [aleph]]0. This provides a simple way of giving a negative answer to a question of E. Weinberg that asks whether or not there is a torsion free group G of cardinality greater than the continuum in which each pure subgroup is indecomposable. Also, a torsion free group G is constructed with the property that G contains maximal pure Independent subsets S and T where T is infinite and S contains only one element. This yields a counterexample to a theorem of Chase. In the remainder of the first chapter, characterizations of purely Indecomposable torsion free groups are given. These characterizations are both of a homological and structural nature. The second chapter is devoted to the investigation of locally free groups and group extensions. The basic problem of this chapter is one posed by J. H. C. Whitehead that asks for a characterization of those groups G for which Ext(G,Z) = 0, where Z denotes the infinite cyclic group. Although an answer to Whitehead's question is not given, necessary and sufficient conditions on the structure of a group G are given in order that Ext(G,S) = 0, where S = [sum][lowered [aleph][lowered 0]] Z . These conditions are that G be locally free and that every subgroup of G of countable index contain a direct summand of G of countable index. It is not known whether these conditions are equivalent to the condition that G be free. Also, a new characterization is given for reduced groups G which have the property that Ext(G,Z) is torsion free. Finally, a method of Chase is generalized for constructing non-free, pure subgroups of [Pi][lowered [aleph][lowered 0]] Z . The last chapter deals with transitive and fully transitive primary groups. With the aid of recent results of Hill, Meglbben, and Nunke, it is shown, for a primary group G, that if G/p[raised beta]G is a direct sum of countable groups and if p[rasied beta]G is fully transitive, then G is fully transitive. The same result is established for transitivity except that [beta] is restricted to be a countable ordinal.

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