For any compact abelian group $K$.$$K\cong H_0\times \mathrm{U}(1)^k,$$where $H_0$ is a finite group.
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1$\begingroup$ I'm guessing that this question won't last long in its current form, but let me point out that you at very least need to add some hypotheses. If the scope of $K$ is "compact topological abelian group" then your statement is false. $\endgroup$– RamseyCommented Oct 9, 2012 at 14:34
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$\begingroup$ This is clearly not true, consider for example $\mathbb Z_2^{\mathbb N}$. I am trying to think of some context where this question makes sense. Is $k$ finite? $\endgroup$– Stefan GeschkeCommented Oct 9, 2012 at 15:05
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1$\begingroup$ Presumably he means compact Hausdorff abelian Lie groups, where $k$ is finite. $\endgroup$– Todd TrimbleCommented Oct 9, 2012 at 15:49
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$\begingroup$ Try looking here for instance: hopf.math.purdue.edu/Dwyer-Wilkerson/lie/liegroups.pdf $\endgroup$– Todd TrimbleCommented Oct 9, 2012 at 15:51
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$\begingroup$ Oh sorry, two comments ago, I should have said "he or she". $\endgroup$– Todd TrimbleCommented Oct 9, 2012 at 15:52
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1 Answer
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Any topological group of the form $K\cong H_0\times U(1)^k$ (with $H_0$ finite and $k$ a positive integer) is a closed subgroup of $U(1)^h$ (for some positive integer $h\geq k$). Furthermore, all the closed subgroups of $U(1)^h$ are of this form.
The proof is an easy application of the Pontryagin-Van Kampen duality. In fact, such groups are the duals of the finitely generated Abelian groups (which are quotients of $\mathbb Z^h$). It is well known that such groups are of the form $F\times \mathbb Z^k$ (with $F$ a finite group).
To find a general form for a compact abelian group is as difficult as giving a structure theorem for discrete abelian group (which is known to be quite a difficult, and fairly open, problem in general, even if there are nice results for countable torsion groups).
EDIT: just to answer also to the comment of Stefan Geschke. Finitely generated groups in $\mathrm{Mod}(\mathbb Z)$ can be characterized as the Noetherian objects of the category. So I guess that (by duality) the objects of the form $K\cong H_0\times U(1)^k$ should be the Artinian objects in the category of compact abelian topological ($T_2$) groups. (N.B.= here by Artinian I mean the category-theoretical notion)
