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

Every finite group, by an elementary counting argument.
More generally, every polycyclic-by-finite group.
Any finitely generated free group.
The additive group Q of rationals.
Any finitely generated residually finite group.
Any word-hyperbolic group.

Examples of non-Hopfian groups

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.