Q:

An ascending chain of subgroups

Let $G$ be the restricted wreath product of two infinite cyclic groups, say $\langle g\rangle$ and $\langle x\rangle$. Say $B$ the base group of $G$ (suppose $x\in B$). Is it possible to find a strictly ascending infinite chain of subgroups which is not contained in the base group?

A:

According to 4.2.3 of this book by Lennox and Robinson, if $G$ is a virtually polycyclic group, then the integral groupring ${\mathbb Z}G$ is Noetherian as a right module. Applying this result with $G$ an infinite cyclic group can be used to prove that there is no such ascending chain of subgroups in the restricted wreath product.

