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I would like to know if there's any structure theorem classifying all compact abelian groups, any reference is appreciated.

I do know there is a structure theorem for compact abelian Lie groups, and for locally compact abelian groups, however the first one is too specific and the second one is too general.

I know there are pro-finite groups, and Lie groups. Recently i discovered there are also Pro-Lie groups (which are not necessarily compact). Is every compact abelian group a direct product of these groups? If so, what can we say about compact Pro-Lie groups?

Thanks!

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    $\begingroup$ Because of Pontryagin duality, classifying all compact abelian groups is the same as classifying (abstract) abelian groups. $\endgroup$ – egreg Jun 26 '17 at 20:17
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    $\begingroup$ Is there any classifications of this groups? $\endgroup$ – Yanko Jun 26 '17 at 20:26
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    $\begingroup$ No; it's a huge problem. $\endgroup$ – egreg Jun 26 '17 at 20:34
  • $\begingroup$ RIght, thanks! Then I guess that is why it is so hard to find a structure theorem for this groups. $\endgroup$ – Yanko Jun 26 '17 at 20:46
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    $\begingroup$ One fact that impressed me when learning about the structure of abelian groups and LCA groups: there are countable abelian groups G such that G is not isomorphic to a group of the form T x H, where T is a torsion group and H is torsion free (See Fuchs's book Infinite Abelian Groups, Volume 1). Pontryagin duality then produces an example of a compact metric abelian group which cannot be written as a product of a connected group and a totally disconnected group. So in general there's nothing as nice as the structure theorem for finitely generated abelian groups. $\endgroup$ – John Griesmer Jun 28 '17 at 1:04
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You can start from the Wikipedia page on Pontryagin duality which also has a few references on classical books on the subject, to which you may add Loth's and Morris' books.

Pontryagin duality can be defined on the locally compact abelian groups, but restricts to a duality between discrete abelian groups and compact ones, so classifying compact abelian groups is the same as classifying (abstract) abelian groups, which is a huge problem, very far from being solved.

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