Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients.
[1] A group G is co-Hopfian if every monomorphism is an isomorphism.
It was shown by Collins (1969) that it is an undecidable problem to determine, given a finite presentation of a group, whether the group is Hopfian.
Unlike the undecidability of many properties of groups this is not a consequence of the Adian–Rabin theorem, because Hopficity is not a Markov property, as was shown by Miller & Schupp (1971).
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