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
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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
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 |