Hopfian group

In mathematics, a Hopfian group is a group G for which every epimorphism


 * G → G

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


 * G → G

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

 * Quasicyclic groups.
 * The additive group R of real numbers.
 * The Baumslag–Solitar group B(2,3). (In general B(m, n) is non-Hopfian if and only if there exists primes p, q with p|m, q|n and p ∤ n, q ∤ m)

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.