Quasi-minimal abelian groups |
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Authors: | B Goldsmith S Ó hó gá in S Wallutis |
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Institution: | School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland ; School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland ; Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany |
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Abstract: | An abelian group is said to be quasi-minimal (purely quasi-minimal, directly quasi-minimal) if it is isomorphic to all its subgroups (pure subgroups, direct summands, respectively) of the same cardinality as . Obviously quasi-minimality implies pure quasi-minimality which in turn implies direct quasi-minimality, but we show that neither converse implication holds. We obtain a complete characterisation of quasi-minimal groups. In the purely quasi-minimal case, assuming GCH, a complete characterisation is also established. An independence result is proved for directly quasi-minimal groups. |
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