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.Examples of Hopfian groups
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.
