Hopfian group

In mathematics, a Hopfian group is a group "G" for which every epimorphism
is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A group "G" is co-Hopfian if every monomorphism
is an isomorphism. Equivalently, "G" is not isomorphic to any of its proper subgroups.
Properties.
It was shown by 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 .