Hopfian group

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In mathematics, a Hopfian group is a group G for which every epimorphism

GG

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

GG

is an isomorphism. Equivalently, G is not isomorphic to any of its proper subgroups.

Examples of Hopfian groups[edit]

Examples of non-Hopfian groups[edit]

Properties[edit]

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[edit]

External links[edit]