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
- 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
- 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.
- 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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