Finitely generated Hopfian group

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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and Hopfian group
View other group property conjunctions OR view all group properties

Definition

A finitely generated Hopfian group is a group that is both a finitely generated group (i.e., it has a finite generating set) and a Hopfian group (i.e., every surjective 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 Finitely generated profinite group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Noetherian group|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
Finitely generated free group finitely generated and a free group; equivalently, free on a finite freely generating set finitely generated and free implies Hopfian Finitely generated residually finite group, Finitely presented conjugacy-separable group, Finitely presented residually finite group|FULL LIST, MORE INFO
Finitely generated abelian group finitely generated and an abelian group finitely generated abelian implies Hopfian 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
Slender group every subgroup is a finitely generated group slender implies Hopfian, (slender implies finitely generated by definition) |FULL LIST, MORE INFO
Finitely generated simple group finitely generated and a simple group simple implies Hopfian |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Hopfian group |FULL LIST, MORE INFO
Finitely generated group |FULL LIST, MORE INFO