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# Hopfian group

A group is said to be *Hopfian* if it is not isomorphic to any of its proper quotients.
A group $G$ is Hopfian if and only if every surjective endomorphism $G\to G$ is an automorphism.

A group is said to be *co-Hopfian* if it is not isomorphic to any of its proper subgroups.
A group $G$ is co-Hopfian if and only if every injective endomorphism $G\to G$ is an automorphism.

# Examples

Every finite group is obviously Hopfian and co-Hopfian.

The group of integers is Hopfian, but not co-Hopfian. More generally, every finitely generated abelian group is Hopfian, but is not co-Hopfian unless it is finite.

Quasicyclic groups are co-Hopfian, but not Hopfian.

Free groups of infinite rank are neither Hopfian nor co-Hopfian. By contrast, free groups of finite rank are Hopfian (though not co-Hopfian unless of rank zero).

By a theorem of Mal’cev, every finitely generated residually finite group is Hopfian.

The Baumslag-Solitar group with presentation $\langle b,t\mid t^{{-1}}b^{2}t=b^{3}\rangle$ is an example of a finitely generated group that is not Hopfian.

## Mathematics Subject Classification

20F99*no label found*

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