I was analyzing the following example of wreath product of groups.
Let $\mathbb{Z}_2$ be the cyclic group of order two and $\mathbb{Z}$ be the usual additive group of integers. Consider the restricted wreath product $G:=\mathbb{Z}_2\wr \mathbb{Z}$ (where $\mathbb{Z}$ acts on $\mathbb{Z}_2$).
The problem I have is the following one: Is it possible to find a (strictly) ascending chain in $G$ of finitely generated non-Abelian subgroups?
I think this is not possible but I cannot show why. Any ideas?