Hopfian group
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.
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 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 |