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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?

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

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    $\begingroup$ The subgroup generated by $x$ is not normal! $\endgroup$ – Steve D May 15 '11 at 20:08
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    $\begingroup$ But the normal subgroup generated by $x$ is normal and infinitely generated. It is isomorphic to the additive group of diadic rational numbers. $\endgroup$ – Mark Sapir May 16 '11 at 5:59
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There are finitely presented metabelian groups containing the lamplighter groups. One of them was constructed by Baumslag: $\langle a,b,c \mid a^2=1, [b,c]=1, [a^b,a]=1, a^c=a^ba\rangle$.

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By the way, you may enjoy the fact, due to G. Baumslag, that a standard wreath product $W\wr G$ with $W\neq 1$ and $G$ infinite, is never finitely presented; see Gilbert Baumslag. Wreath products of finitely presented groups. Math. Z. 75 , 22-28, 1961. For finite presentability of permutational wreath products, see a paper by Cornulier: http://www.normalesup.org/~cornulier/wrea_fp.pdf

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