In mathematics, a Hopfian group is a group G for which every surjective homomorphism 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 injective homomorphism G → G is an isomorphism. Equivalently, G is not isomorphic to any of its proper subgroups.
Hopfian groups are named after Heinz Hopf.
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 of rationals. Any finitely generated residually finite group. Any word-hyperbolic group.
Examples of non-Hopfian groups Quasicyclic groups. The additive group of real numbers. The Baumslag–Solitar group . In general is non-Hopfian if and only if there exists primes and with , and , .
Properties
It was shown by Collins 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.
Notes
References
External links
Non-Hopf group in the Encyclopedia of Mathematics