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The notions of Hopfian and co-Hopfian groups have been of interest for some time. In this present work we characterize the more restricted classes of hereditarily Hopfian (co-Hopfian) and super Hopfian (co-Hopfian) groups in the case where the groups are Abelian.
Goldsmith, B. & Gong, K. (2012). On super and hereditarily hopfian and co-hopfian Abelian groups. Archiv der Mathematik, 99(1), pp.1-8.. doi:10.1007/s00013-012-0402-2