# Finitely generated and nilpotent implies Hopfian

From Groupprops

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 nilpotent group) must also satisfy the second group property (i.e., Hopfian group)

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## Statement

Any finitely generated nilpotent group is a Hopfian group.

## Facts used

- Equivalence of definitions of finitely generated nilpotent group: In particular, the equivalence of interest is that for a nilpotent group, being finitely generated is equivalent to being Noetherian (i.e., every subgroup is finitely generated).
- Noetherian implies Hopfian

## Proof

The proof follows directly by combining Facts (1) and (2).