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

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## Statement

Any Noetherian group (i.e., a group in which every subgroup is finitely generated) is Hopfian (i.e., it is not isomorphic to the quotient by any nontrivial subgroup).

## Definitions used

### Noetherian group

`Further information: Noetherian group`

A group is termed Noetherian if every subgroup of the group is finitely generated. Equivalently, it satisfies the ascending chain condition of subgroups: every ascending chain of subgroups stabilizes after a finite length.

### Hopfian group

`Further information: Hopfian group`

A group is termed Hopfian if it is not isomorphic to its quotient by any nontrivial normal subgroup.

## Facts used

- Noetherian implies ascending chain condition on normal subgroups
- Ascending chain condition on normal subgroups implies Hopfian

## Proof

The proof follows by piecing together facts (1) and (2).