Hopfian group


finite rank

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

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


Every finite groupMathworldPlanetmath 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 generatedMathworldPlanetmathPlanetmathPlanetmath abelian groupMathworldPlanetmath is Hopfian, but is not co-Hopfian unless it is finite.

Quasicyclic groups are co-Hopfian, but not Hopfian.

Free groupsMathworldPlanetmath 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 presentationMathworldPlanetmathPlanetmathPlanetmath b,tt-1b2t=b3 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