Explore: Hopfian Groups

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

examples of non-Hopfian groups. The groups contain residually finite groups, Hopfian groups that are not residually finite, and non-Hopfian groups. Define A

of group theory, a co-Hopfian group is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named

honor of Heinz Hopf and his use of the concept of the hopfian group in his work on fundamental groups of surfaces. (Hazewinkel 2001, p. 63) Both conditions

pure mathematics. Co-Hopfian group Cohomotopy group EHP spectral sequence Hopfian group Hopfian object Hopf algebra Quantum group Hopf fibration Alexandroff

non-residually finite groups can be constructed using the fact that all finitely generated residually finite groups are Hopfian groups. For example the Baumslag–Solitar

finitely generated one-relator group that is not Hopfian and therefore not residually finite, for example the Baumslag–Solitar group B ( 2 , 3 ) = ⟨ a , b ∣

{\displaystyle \pi _{1}} . An argument for higher homotopy groups remains open. Also there are non-Hopfian groups. Another famous question of Hopf is the Hopf product

Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society 68 (1962), 199–201. MR 0142635

the property of being Hopfian is undecidable for finitely presentable groups, while neither being Hopfian nor being non-Hopfian are Markov. Higman's embedding