A group is said to be Hopfian if it is not isomorphic to any of its proper quotients (http://planetmath.org/QuotientGroup). A group is Hopfian if and only if every surjective endomorphism is an automorphism.
A group is said to be co-Hopfian if it is not isomorphic to any of its proper subgroups. A group is co-Hopfian if and only if every injective endomorphism is an automorphism.
Every finite group is obviously Hopfian and co-Hopfian.
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.
|Date of creation||2013-03-22 15:36:03|
|Last modified on||2013-03-22 15:36:03|
|Last modified by||yark (2760)|