I'm looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated.


The question is easy for finitely generated amenable group and an example is the lamp-lighter group $C\_2\wr \mathbb{Z}$. 


An Abelian and finitely generated group has no such subgroups. There exists a bigger class of groups with this property? 



Example of an amenable finitely generated and presented group with a non-finitely generated subgroup
I don't know much about amenable groups I am afraid, but according to the Wikipedia article, all solvable groups are amenable. So we can take the Baumslag-Solitar group


$B(1,n) = \langle x,y \mid y^{-1}xy = x^n \rangle.$


If we let $N$ be the normal closure of the subgroup generated by $x$, then $N$ is abelian with $G/N$ cyclic, but $N$ is not finitely generated when $n > 1$. Note also that $B(1,n)$ is isomorphic to the subgroup of ${\rm GL}(2, \mathbb{Q})$ generated by


$x = \left(\begin{array}{cc}1&0\\\\1&1\end{array}\right)$ and
$y = \left(\begin{array}{cc}n&0\\\\0&1\end{array}\right).$


