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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 after...

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...

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...

in the field of pure mathematics. Co-Hopfian group Cohomotopy group EHP spectral sequence Hopfian group Hopfian object Hopf algebra Quantum group Hopf...

generated, residually finite, and Hopfian , but B ( 2 , 3 ) {\displaystyle B(2,3)} is finitely generated yet not Hopfian, and therefore not residually finite...

contradicting 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...

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

its image is a cyclic subgroup of G. In particular the group G is Hopfian and co-Hopfian. The outer automorphism group Out(G) of G is isomorphic to the additive...

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

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...

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

F(X). There exists a finitely generated one-relator group that is not Hopfian and therefore not residually finite, for example the Baumslag–Solitar group...

tree techniques to prove that torsion-free word-hyperbolic groups are Hopfian. This result and Sela's approach were later generalized by others to finitely...