Hopfian group
From Wikipedia, the free encyclopedia
In mathematics, a Hopfian group is a group G for which every epimorphism
- G → G
is an isomorphism. A group is Hopfian if and only if it is not isomorphic to any of its proper quotients.
Every finite group is Hopfian for elementary reasons. More generally, every polycyclic-by-finite group is Hopfian. The group Q of rationals is also Hopfian.
Quasicyclic groups are not Hopfian, nor is the group R of real numbers. For more complicated examples of non-Hopfian groups see Baumslag-Solitar group.
