# Hopfian group

finite rank

A group is said to be Hopfian if it is not isomorphic to any of its proper quotients (http://planetmath.org/QuotientGroup). A group $G$ is Hopfian if and only if every surjective endomorphism $G\to G$ is an automorphism.

A group is said to be co-Hopfian if it is not isomorphic to any of its proper subgroups. A group $G$ is co-Hopfian if and only if every injective endomorphism $G\to G$ is an automorphism.

## Examples

Every finite group is obviously Hopfian and co-Hopfian.

The group of rationals is an example of an infinite group that is both Hopfian and co-Hopfian.

The group of integers is Hopfian, but not co-Hopfian. More generally, every finitely generated abelian group is Hopfian, but is not co-Hopfian unless it is finite.

Quasicyclic groups are co-Hopfian, but not Hopfian.

Free groups of infinite rank are neither Hopfian nor co-Hopfian. By contrast, free groups of finite rank are Hopfian (though not co-Hopfian unless of rank zero).

By a theorem of Mal’cev, every finitely generated residually finite group is Hopfian.

The Baumslag-Solitar group with presentation $\langle b,t\mid t^{-1}b^{2}t=b^{3}\rangle$ is an example of a finitely generated group that is not Hopfian.

 Title Hopfian group Canonical name HopfianGroup Date of creation 2013-03-22 15:36:03 Last modified on 2013-03-22 15:36:03 Owner yark (2760) Last modified by yark (2760) Numerical id 11 Author yark (2760) Entry type Definition Classification msc 20F99 Related topic HopfianModule Defines Hopfian Defines co-Hopfian Defines cohopfian Defines co-Hopfian group Defines cohopfian group