Hopfian group

In mathematics, a Hopfian group is a group G for which every surjective homomorphism G → G is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients.Template:Sfnp

A group G is co-Hopfian if every injective homomorphism G → G is an isomorphism. Equivalently, G is not isomorphic to any of its proper subgroups.

Hopfian groups are named after Heinz Hopf.

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 $\mathbb{Q}$ 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.Template:Sfnp
• The Baumslag–Solitar group $B(2,3)$. In general $B(m,n)$ is non-Hopfian if and only if there exists primes $p$ and $q$ with $p\mid m$, $q\mid n$ and $p\nmid n$, $q\nmid m$.Template:Sfnp

Properties

It was shown by Collins that it is an undecidable problem to determine, given a finite presentation of a group, whether the group is Hopfian.Template:Sfnp 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.Template:Sfnp

References

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External links

• Template:PlanetMath
• Non-Hopf group in the Encyclopedia of Mathematics

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