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This thesis investigates those abelian groups which are minimal with respect to certain quasi-orders defined on Abk, the category of abelian groups of a given infinite cardinality k. Six such quasi-orders are defined and groups which are minimal with respect to these quasi-orders are called either quasi-minimal, with the associated concepts of purely and directly quasi-minimal groups, or simple minimal with the corresponding associated groups. A complete characterisation is derived for the quasi-minimal groups and, assuming GCH, for the purely quasi-minimal groups. Moreover, it is shown that the direct quasi-minimality of a group may be undecidable in ZFC. In the minimal case, consideration of torsion groups can be reduced to that of p-groups, and a criterion for the minimality of a p-group is found in terms of its Ulm invariants. The minimality of various classes of torsion free groups is determined. In particular, a characterisation in terms of their critical typesets is found for all finite rank and for large classes of infinite rank completely decomposable groups. Sever equivalent conditions are given for the minimality of general separable groups. The minimality of mixed groups is also investigate, particularly those of torsion-free rank 1.
O hOgain, Seosamh. Aspects of minimality in Abelian groups. Dublin : Dublin Institute of Technology, 2001.