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 nonimplications Group metaproperty satisfactions  Group metaproperty dissatisfactions  Group property satisfactions  Group property dissatisfactions
This is a variation of finite groupFind 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:
Metaproperties
Relation with other properties
Stronger properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

finite group 
order is finite 


Finitely generated Hopfian group, Finitely generated profinite group, Finitely generated residually finite group, Finitely presented conjugacyseparable group, Group satisfying ascending chain condition on normal subgroups, Noetherian groupFULL LIST, MORE INFO

finitely generated free group 
free group on finite generating set 
finitely generated and free implies Hopfian 

Finitely generated Hopfian group, Finitely generated residually finite group, Finitely presented conjugacyseparable group, Finitely presented residually finite groupFULL LIST, MORE INFO

finitely generated residually finite group 
finitely generated and residually finite 
finitely generated and residually finite implies Hopfian 

Finitely generated Hopfian groupFULL 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) 

Finitely generated Hopfian group, Group satisfying ascending chain condition on normal subgroups, Group satisfying ascending chain condition on subnormal subgroupsFULL 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 

Group in which every endomorphism is trivial or injective, Group satisfying ascending chain condition on normal subgroups, Group satisfying ascending chain condition on subnormal subgroupsFULL 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 

Group in which every endomorphism is trivial or injectiveFULL 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 nonabelian group 
Finitely generated Hopfian group, Finitely generated conjugacyseparable group, Finitely generated residually finite group, Finitely presented conjugacyseparable group, Finitely presented residually finite group, Noetherian groupFULL LIST, MORE INFO

finitely generated nilpotent group 

finitely generated and nilpotent implies Hopfian (via Noetherian) 
(via Noetherian, also any finite nonnilpotent counterexample) 
Noetherian groupFULL LIST, MORE INFO

Incomparable properties