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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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

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 \le 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 Finitely generated Hopfian group, Finitely generated profinite group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Group satisfying ascending chain condition on normal subgroups, Noetherian group|FULL 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 conjugacy-separable group, Finitely presented residually finite group|FULL LIST, MORE INFO
finitely generated residually finite group finitely generated and residually finite finitely generated and residually finite implies Hopfian Finitely generated Hopfian group|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) Finitely generated Hopfian group, Group satisfying ascending chain condition on normal subgroups, Group satisfying ascending chain condition on subnormal 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 Group in which every endomorphism is trivial or injective, Group satisfying ascending chain condition on normal subgroups, Group satisfying ascending chain condition on subnormal subgroups|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 Group in which every endomorphism is trivial or injective|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 Finitely generated Hopfian group, Finitely generated conjugacy-separable group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Finitely presented residually finite group, Noetherian 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) Noetherian group|FULL LIST, MORE INFO

Incomparable properties