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The notions of Hopfian and co-Hopfian groups have been of interest for some time. In this present work we exploit some unpublished ideas of Corner to answer questions relating to such groups. In particular, we extend an answer given by Corner to a problem of Beaumont and Pierce and show how the properties may be lifted from subgroups to the whole group in certain situations.
Goldsmith, B. & Gong, K. (2012). A Note on Hopfian and Co-Hopfian Abelian Groups. Proceedings of the American Mathematical Society, vol. 576, pg. 129–136. doi:10.1090/conm/576/11356.