Finitely generated not implies Noetherian

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.

A general construction using a restricted wreath product
Let $$A$$ be a nontrivial finitely generated group. Let $$G$$ be the uses as intermediate construct::restricted external wreath product of $$A$$ and the uses as intermediate construct::group of integers $$\mathbb{Z}$$ acting regularly. In other words, $$G$$ is the uses as intermediate construct::external semidirect product of $$H$$ and $$\mathbb{Z}$$, where $$H$$ is the uses as intermediate construct::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 $$H$$.