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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!

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up vote 3 down vote accepted

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

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

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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. –  Arturo Magidin Jun 12 '12 at 21:31
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