# nLab Hopfian group

A discrete group $G$ is a Hopfian group if every surjective endomorphism $\phi : G\to G$ is an isomorphism. Dually, a discrete group is called coHopfian if any injective endomorphism of $G$ is an isomorphism.

As the epimorphisms and monomorphisms in Grp are precisely the surjections and injections, the definition generalises immediately to that of a Hopfian object? in any category. (But perhaps one might want to require regular epimorphisms, or …?)

Clearly all finite groups are both Hopfian and coHopfian. Using Nielsen’s method, one can show that every finitely generated free group and the union of any ascending chain of such free groups are Hopfian. It is also known that every torsion-free hyperbolic group? is Hopfian.

Revised on August 9, 2009 16:28:20 by Toby Bartels (71.104.230.172)