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.

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 ${\displaystyle \mathbb {Q} }$ of rationals.
• Any finitely generated residually finite group.
• Any word-hyperbolic group.

Examples of non-Hopfian groups

• Quasicyclic groups.
• The additive group ${\displaystyle \mathbb {R} }$ of real numbers.
• The Baumslag–Solitar group ${\displaystyle B(2,3)}$. In general ${\displaystyle B(m,n)}$ is non-Hopfian if and only if there exists primes ${\displaystyle p}$ and ${\displaystyle q}$ with ${\displaystyle p\mid m}$, ${\displaystyle q\mid n}$ and ${\displaystyle p\nmid n}$, ${\displaystyle q\nmid m}$.

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. 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.

Notes

References

External links

• Hopfian group at PlanetMath.
• Non-Hopf group in the Encyclopedia of Mathematics

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