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...

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...

In the branch of mathematics called category theory, a hopfian object is an object A such that any epimorphism of A onto A is necessarily an automorphism...

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

field of pure mathematics. Co-Hopfian group Cohomotopy group EHP spectral sequence Hopfian group Hopfian object Quantum group Alexandroff P., Hopf H. Topologie...

Baumslag–Solitar group B(2,3) is not Hopfian, and therefore not residually finite. Every group G may be made into a topological group by taking as a basis...

the degree-1 assumption. This implies that the conjecture holds for Hopfian groups, as for them one then gets that f ∗ {\displaystyle f_{*}} is an isomorphism...

Gilbert; Solitar, Donald (1962), "Some two-generator one-relator non-Hopfian groups", Bulletin of the American Mathematical Society, 68 (3): 199–201, doi:10...

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 ∣...

In particular the group G is Hopfian and co-Hopfian. The outer automorphism group Out(G) of G is isomorphic to the additive group of dyadic rationals...

Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society 68 (1962), 199–201. MR0142635...

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

automorphism of M. This says simply that M is a Hopfian module. Similarly, an Artinian module M is coHopfian: any injective endomorphism f is also a surjective...

word-hyperbolic groups are Hopfian. This result and Sela's approach were later generalized by others to finitely generated subgroups of hyperbolic groups and to...