Hopfian group
From Wikipedia, the free encyclopedia
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.
- 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]
