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Hopfian group

From Wikipedia, the free encyclopedia

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.[1]

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

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Examples of non-Hopfian groups

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Properties

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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

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  1. ^ Florian Bouyer. "Definition 7.6.". Presentation of Groups (PDF). University of Warwick. A group G is non-Hopfian if there exists 1 ≠ N ◃ G such that G/N ≅ G
  2. ^ Clark, Pete L. (Feb 17, 2012). "Can you always find a surjective endomorphism of groups such that it is not injective?". Math Stack Exchange. This is because (R,+) is torsion-free and divisible and thus a Q-vector space. So -- since every vector space has a basis, by the Axiom of Choice -- it is isomorphic to the direct sum of copies of (Q,+) indexed by a set of continuum cardinality. This makes the Hopfian property clear.
  3. ^ Florian Bouyer. "Theorem 7.7.". Presentation of Groups (PDF). University of Warwick.
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