Hopfian group

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.

Example of Hopfian groups

* Every finite group, by an elementary counting argument.
* More generally, every polycyclic-by-finite group.
* Any finitely-generated free group.
* The group Q of rationals.

Examples of non-Hopfian groups

* Quasicyclic groups.
* The group R of real numbers.
* The Baumslag-Solitar group "B"(2,3).

References

*

External links

* [http://planetmath.org/encyclopedia/HopfianGroup.html PlanetMath page]
* [http://eom.springer.de/N/n067060.htm EoM page]


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