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Aspects Of Minimality In Abelian Groups, Seosamh Ó Hógáin
Aspects Of Minimality In Abelian Groups, Seosamh Ó Hógáin
Doctoral
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 …