Hopfian group

In mathematics, a Hopfian group is a group G for which every surjective homomorphism G → G 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 injective homomorphism G → G is an isomorphism. Equivalently, G is not isomorphic to any of its proper subgroups.

Hopfian groups are named after Heinz Hopf.

Examples of Hopfian groups

Examples of non-Hopfian groups

  • Quasicyclic groups.
  • The additive group of real numbers.[2]
  • The Baumslag–Solitar group . In general is non-Hopfian if and only if there exists primes and with , and , .[3]

Properties

It was shown by Collins that it is an undecidable problem to determine, given a finite presentation of a group, whether the group is Hopfian.[4] 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.[5]

Notes

  1. ↑ Bouyer (2013), definition 7.6.
  2. ↑ Clark (2012), "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. ↑ Bouyer (2013), theorem 7.7.
  4. ↑ Collins (1969).
  5. ↑ Miller & Schupp (1971).

References

  • Bouyer, Florian (2013). Presentation of Groups (PDF). University of Warwick.
  • Johnson, D. L. (1990) [1976]. Presentations of Groups. London Mathematical Society Student Texts. Vol. 15 (2nd ed.). Cambridge University Press. p. 35. ISBN 978-0-52158542-2.


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