# 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?

• Do you mean the restricted wreath product (i.e. only finitely many nontrivial components in the base group) or the full wreath product? For the restricted one I think the answer is no, but it is certainly yes for the unrestricted version, since you can find an infinite chain of finitely generated subgroups in an uncountable group. – Derek Holt Sep 22 '16 at 17:52
• I mean restricted, infact I also thought the answer was no, but I cannot find a good idea to prove it. – W4cc0 Sep 23 '16 at 6:54

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.