# Finitely generated not implies Noetherian

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This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated group) neednotsatisfy the second group property (i.e., slender group)

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This article gives the statement, and possibly proof, of a group property (i.e., finitely generated group)notsatisfying a group metaproperty (i.e., subgroup-closed group property).

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## Statement

The statement has the following equivalent formulations:

- A subgroup of a finitely generated group need not be finitely generated, i.e., the property of being finitely generated is not a subgroup-closed group property.
- A finitely generated group need not be a slender group, i.e., it may not be true that every subgroup of the group is also finitely generated.

## Proof

### A general construction using a restricted wreath product

Let be a nontrivial finitely generated group. Let be the restricted external wreath product of and the group of integers acting regularly. In other words, is the external semidirect product of and , where is the restricted external direct product of countably many copies of and acts on the coordinates by a shift of one.

Now, we see that:

- is finitely generated: In fact, is generated by a generating set for the in any one coordinate and a generator for the that does the coordinate shifts.
- is not finitely generated: Any finite subset of has a total of only finitely many nontrivial coordinates, hence it cannot generate the whole group .