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An abelian group $G$ 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 $G$. 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.
Brendan Goldsmith , O hOgain, S. and S. Wallutis:Quasi-minimal Abelian groups. Proceedings of the American Mathematical Society, Vol. 132, 8 (2004), pp.2185-2195. doi.org/10.21427/9x45-c406