Hopfian group

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 properties
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 finite group|Find other variations of finite group |

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

Definition

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

• It is not isomorphic to the quotient group by any nontrivial normal subgroup (in short, it is not isomorphic to any of its proper quotients).
• Every surjective endomorphism of it is an automorphism.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	No	Hopfianness is not subgroup-closed	It is possible to have a Hopfian group $G$ and a subgroup $H\leq G$ such that $H$ is not Hopfian.
quotient-closed group property	No	Hopfianness is not quotient-closed	It is possible to have a Hopfian group $G$ and a normal subgroup $H$ of $G$ such that the quotient group $G/H$ is not Hopfian.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite group	order is finite			|FULL LIST, MORE INFO
finitely generated free group	free group on finite generating set	finitely generated and free implies Hopfian		|FULL LIST, MORE INFO
finitely generated residually finite group	finitely generated and residually finite	finitely generated and residually finite implies Hopfian		|FULL LIST, MORE INFO
group satisfying ascending chain condition on normal subgroups	there is no infinite strictly ascending chain of normal subgroups	ascending chain condition on normal subgroups implies Hopfian		|FULL LIST, MORE INFO
Noetherian group (also called slender group)	every subgroup is finitely generated; equivalently, no infinite strictly ascending chain of subgroups	Noetherian implies Hopfian (proof is ascending chain condition on normal subgroups)		|FULL LIST, MORE INFO
finitely generated Hopfian group	finitely generated and Hopfian	(by definition)	Hopfian not implies finitely generated	|FULL LIST, MORE INFO
simple group	has no proper nontrivial normal subgroups	simple implies Hopfian		|FULL LIST, MORE INFO
group in which every endomorphism is trivial or an automorphism	every endomorphism is either trivial or is an automorphism	every endomorphism is trivial or an automorphism implies Hopfian		|FULL LIST, MORE INFO
Group in which every endomorphism is trivial or injective	every endomorphism is either trivial or an injective endomorphism	every endomorphism is trivial or injective implies Hopfian	(finite counterexamples)	|FULL LIST, MORE INFO
finitely generated abelian group		finitely generated abelian implies Hopfian (also via Noetherian)	any finite non-abelian group	|FULL LIST, MORE INFO
finitely generated nilpotent group		finitely generated and nilpotent implies Hopfian (via Noetherian)	(via Noetherian, also any finite non-nilpotent counterexample)	|FULL LIST, MORE INFO

Incomparable properties

• Co-Hopfian group