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) need not satisfy 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) not satisfying a group metaproperty (i.e., subgroup-closed group property).
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Statement

The statement has the following equivalent formulations:

  1. 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.
  2. 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 A be a nontrivial finitely generated group. Let G be the restricted external wreath product of A and the group of integers \mathbb{Z} acting regularly. In other words, G is the external semidirect product of H and \mathbb{Z}, where H is the restricted external direct product of countably many copies of A and \mathbb{Z} acts on the coordinates by a shift of one.

Now, we see that:

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