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.^[1]

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.^[2]
• 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)^[3]

Properties

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).