Supriya Ghosh (Editor)

Hopfian group

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

Contents

G → G

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

G → G

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

Examples of Hopfian groups

  • Every finite group, by an elementary counting argument.
  • More generally, every polycyclic-by-finite group.
  • Any finitely-generated free group.
  • The group Q of rationals.
  • Any finitely generated residually finite group.
  • Any torsion-free word-hyperbolic group.

    • Examples of non-Hopfian groups

  • Quasicyclic groups.
  • The group R of real numbers.
  • The Baumslag–Solitar group B(2,3).

    • Properties

    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

    Hopfian group Wikipedia
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