1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Derek Holt Sep 22 '16 at 17:52
  • $\begingroup$ I mean restricted, infact I also thought the answer was no, but I cannot find a good idea to prove it. $\endgroup$ – W4cc0 Sep 23 '16 at 6:54
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.