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Let $X$ be a torsion-free, totally disconnected, locally compact abelian group. I want to find a non-zero subgroup of $X$ with the following conditions:

1- compact

2- open

3- $n-$divisible for some $n$.

Of course, it is clear that $\bigcap_{r=1}^{\infty}n^{r}K$ is $n-$divisible and compact, for some compact open subgroup $K$ of $X$ and $n$. But it need not be non-zero.

I will be grateful if you can help.

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  • $\begingroup$ Why do you need such a subgroup? To quotient it out? $\endgroup$ Aug 5, 2013 at 13:05
  • $\begingroup$ It does not always exist (e.g., when $X$ is discrete), as it has already been noticed. The question should have been edited accordingly. $\endgroup$
    – YCor
    Apr 28, 2019 at 12:07

1 Answer 1

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Below $X$ is a totally disconnected locally compact Hausdorff abelian group.

If $X$ is $\mathbb Z$ with discrete topology then the only compact subgroup of $X$ is the zero subgroup. So I suggest to interpret the largeness of the searched compact group not as “non-zero”, but as open.

If $\mathcal L$ in Proposition 3.3.9 from [DPS] denotes the class of all locally compact Hausdorff abelian groups then $X$ has a linear topology and contains an open compact subgroup.

An abelian totally disconnected compact (even countably compact) Hausdorff topological group $X$ is reduced, that is $X$ has no non-zero divisible subgroups [AT, Pr. 9.12.A].

Proposition 3.5.9 from [DPS] describes the structure of totally disconnected compact abelian groups.

At last I remark that the group $\mathbb Z_p$ of p-adic integers should be torsion free and $n$-divisible for each $n$ coprime with $p$ (see, for instance, [M, p.4]).

Good luck!

References

[AT] Alexander V. Arhangel'skii, Mikhail G. Tkachenko, Topological groups and related structures, Atlantis Press, Paris; World Sci. Publ., NJ, 2008.

[DPS] Dikran N. Dikranjan, Ivan R. Prodanov, Luchezar N. Stoyanov. Topological Groups, Marcel Dekker, New-York, 1990.

[M] David A. Madore, A first introduction to p-adic numbers.

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    $\begingroup$ More precisely, the conclusion is that every torsion-free, totally disconnected locally compact abelian group has a compact open subgroup of the form $\prod_p\mathbf{Z}_p^{\alpha_p}$, where $p$ ranges over primes and $\alpha_p$ is a cardinal, and $\mathbf{Z}_p$ the $p$-adic group. Moreover, every compact open subgroup has this form, and $\alpha_p$ does not depend on the choice of compact open subgroup. $\endgroup$
    – YCor
    Apr 28, 2019 at 12:11
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    $\begingroup$ In particular, there exists a compact open subgroup that is $n$-divisible for some $n\ge 2$ if and only if $\alpha_p=0$ for some $p$. For instance, if $G$ itself is the profinite completion $\widehat{\mathbf{Z}}$, then $G$ is isomorphic to each of its open subgroups, none of which is $n$-divisible for any $n\ge 2$. $\endgroup$
    – YCor
    Apr 28, 2019 at 12:12

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