# 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 $G$ 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^{} $\u27e8b,t\mid {t}^{-1}{b}^{2}t={b}^{3}\u27e9$ 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 |