Finitely generated and residually finite implies Hopfian
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated residually finite group) must also satisfy the second group property (i.e., Hopfian group)
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Statement
Any finitely generated residually finite group (i.e., a group that is both finitely generated and residually finite) is a Hopfian group.
Definitions used
| Term | Definition used here |
|---|---|
| Finitely generated group | has a finite generating set |
| Residually finite group | every non-identity element is outside some normal subgroup of finite index |
| Hopfian group | every surjective endomorphism of the group is an automorphism |
Proof
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