4
$\begingroup$

I was looking through Lang's Algebra and found the following statement,

Let $G$ be a finite group. An abelian tower of $G$ admits a cyclic refinement.

After some work, I understand the proof, and now I want to show that we cannot drop the hypothesis that $G$ is finite. Its enough to find an infinite abelian group which does not have a cyclic tower. Does anyone know of any such groups? Thanks!

1 Answers 1

3

$\large(\mathbb{Q},+){}{}{}{}{}{}{}{}{}$

Also: $\large(\mathbb{Z}_{p^{\infty}},+).$

  • 0
    The latter example assumes you want a *finite* tower: every element of $\mathbb{Z}_{p^{\infty}}$ has finite order, so a finite tower of subgroups would necessarily yield a finite group.2012-06-12