Hopfian group explained

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 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 .

References

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

External links

• • Non-Hopf group in the Encyclopedia of Mathematics

Notes and References

1. Book: Presentation of Groups. Florian Bouyer. University of Warwick. Definition 7.6.. A group G is non-Hopfian if there exists 1 ≠ N ◃ G such that G/N ≅ G.
2. Web site: 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. Book: Presentation of Groups. Florian Bouyer. University of Warwick. Theorem 7.7..