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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 is Hopfian if and only if every surjective endomorphism is an automorphism.
A group is said to be co-Hopfian if it is not isomorphic to any of its proper subgroups. A group is co-Hopfian if and only if every injective endomorphism 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 is an example of a finitely generated group that is not Hopfian.
Mathematics Subject Classification
20F99 None of the above, but in MSC2010 section 20Fxx- Forums
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