Co-Hopfian group

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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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:

• 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