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 conjugacyseparable group, Noetherian groupFULL LIST, MORE INFO

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

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

Finitely generated abelian group 
finitely generated and an abelian group 
finitely generated abelian implies Hopfian 

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

Slender group 
every subgroup is a finitely generated group 
slender implies Hopfian, (slender implies finitely generated by definition) 

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Finitely generated simple group 
finitely generated and a simple group 
simple implies Hopfian 

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Weaker properties