Hopfian group

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

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 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 additive group Q of rationals.
  • Any finitely generated residually finite group.
  • Any word-hyperbolic group.

Examples of non-Hopfian groups

  • Quasicyclic groups.
  • The additive group R of real numbers.[2]
  • The Baumslag–Solitar group B(2,3). (In general B(m, n) is non-Hopfian if and only if there exists primes p, q with p|m, q|n and p ∤ n, q ∤ m)[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

  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.
  • Collins, D. J. (1969). "On recognising Hopf groups". Archiv der Mathematik. 20 (3): 235–240. doi:10.1007/BF01899291. S2CID 119354919.
  • Johnson, D. L. (1990). Presentations of groups. London Mathematical Society Student Texts. Vol. 15. Cambridge University Press. p. 35. ISBN 0-521-37203-8.
  • Miller, C. F.; Schupp, P. E. (1971). "Embeddings into hopfian groups". Journal of Algebra. 17 (2): 171. doi:10.1016/0021-8693(71)90028-7.


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