Finitely generated Hopfian group
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 | |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 | |FULL LIST, MORE INFO | |
| Finitely generated abelian group | finitely generated and an abelian group | finitely generated abelian implies Hopfian | |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 |