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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. 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.

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 group Q of rationals.
Any finitely generated residually finite group.
Any torsion-free word-hyperbolic group.

Examples of non-Hopfian groups

Quasicyclic groups.
The group R of real numbers.
The Baumslag–Solitar group B(2,3).

Properties

It was shown by Collins (1969) 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, as was shown by Miller & Schupp (1971).
References
D. L. Johnson (1990). Presentations of groups. London Mathematical Society Student Texts. 15. Cambridge University Press. p. 35. ISBN 0-521-37203-8.
Collins, D. J. (1969). "On recognising Hopf groups". Archiv der Mathematik. 20 (3): 235. doi:10.1007/BF01899291.
Miller, C. F.; Schupp, P. E. (1971). "Embeddings into hopfian groups". Journal of Algebra. 17 (2): 171. doi:10.1016/0021-8693(71)90028-7.

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