# Co-Hopfian group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

*This property makes sense for infinite groups. For finite groups, it is always true*

## Contents

## Definition

### Symbol-free definition

A group is termed **co-Hopfian** if it satisfies the following equivalent conditions:

- It is not isomorphic to any proper subgroup
- Every injective endomorphism of the group is an automorphism

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Finite group | Artinian group|FULL LIST, MORE INFO | |||

Artinian group | Artinian implies co-Hopfian | co-Hopfian not implies Artinian | |FULL LIST, MORE INFO | |

Group in which every endomorphism is trivial or an automorphism | |FULL LIST, MORE INFO |

### Related properties

Property | Meaning | Proof of one non-implication | Proof of other non-implication | Notions stronger than both | Notions weaker than both |
---|---|---|---|---|---|

Hopfian group | not isomorphic to any proper quotient group; equivalently, every surjective endomorphism is an automorphism | co-Hopfian not implies Hopfian | Hopfian not implies co-Hopfian | Group in which every endomorphism is trivial or an automorphism|FULL LIST, MORE INFO | |FULL LIST, MORE INFO |

Locally finite group | any finitely generated subgroup is finite | co-Hopfian not implies locally finite | locally finite not implies co-Hopfian | |FULL LIST, MORE INFO | |FULL LIST, MORE INFO |

Periodic group | every non-identity element has finite order | co-Hopfian not implies periodic | periodic not implies co-Hopfian | Artinian group|FULL LIST, MORE INFO | |FULL LIST, MORE INFO |