1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Why do you need such a subgroup? To quotient it out? $\endgroup$ – Patrick Da Silva Aug 5 '13 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 at 12:07
1
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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 at 12:11
  • 1
    $\begingroup$ 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 at 12:12

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.