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An abelian group is said to be minimal if it is isomorphic to all its subgroups of finite index. We obtain a complete characterisation of such groups in the torsion case; in the case of mixed groups of rank 1 we obtain a characterisation for some large classes of such groups.
Goldsmith, Brendan and O hOgain, S.: On torsion and mixed minimal Abelian groups. Rocky Mountain Journal of Mathematics, Vol. 32, (2002), pp.1451-1465.