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 Script error: No such module "Footnotes". 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 Script error: No such module "Footnotes"..
References
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