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Chapter 1 summarises some necessary results concerning free, divisible, algebraically compact and cotorsion groups. A detailed proof of the well-known result that the Specker group P is ℵ1 free but not free is included and the structure of the quotient group P/S is determined. The basic properties of slender groups are examined and cotorsion groups of cardinality less than or equal to that of the continuum are shown to be epimorphic images of P. Chapter II presents Nunke’s characterisation of slender groups. This approach establishes that homomorphic images of the Specker group P are the direct sum of a cotorsion group and a product and uses this result to characterise slender groups. Chapter III presents De Marco and Orsatti’s topological characterisation of slender groups; a reduced torsion free group is slender if and only if it cannot be endowed with a metrisable, linear, non-discrete, complete Hausdorff topology. Chapter IV presents the Los-Eda set theoretic characterisation of slender groups using ultrafilters. The background necessary for the definitions of ℵ1-complete ultrafilters and measurable cardinals is explored in some detail. Chapter V demonstrates the equivalence of the Nunke, De Marco and Orsatti and the Los-Eda characterisations of slender groups.
Kelly, T. (1998). Characterisations of slender groups. Masters dissertation. Technological University Dublin. doi:10.21427/D7BS54