# 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
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## 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