Co-Hopfian group

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

Definition

Symbol-free definition

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

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