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Can anyone give me a reference to the aforementioned theorem? W. Hodges uses it for an example in his "Model Theory", but I couldn't find anything on it yet.

The group may be (let's say, countably) infinite, the direct sum will then be infinite, too, of course. The "exponent" of a group is defined here as $\sup_{g \in G} \mathopen|G.g\mathclose|$

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    $\begingroup$ I think this is usually called Prufer's first theorem (I don't know how to do the umlaut), and can be found in any group theory book which does a fair amount on abelian groups: Kaplansky's book, for example. $\endgroup$
    – user641
    Jul 23, 2011 at 20:16
  • $\begingroup$ Solved by Steve D's comment above. Thanks a lot! (can't mark this question solved as I wasn't logged in when I asked it) $\endgroup$
    – srs
    Jul 24, 2011 at 10:22
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    $\begingroup$ The moderators have merged your two accounts. If @Steve would post his answer as an answer, you can accept it. Otherwise, if would also be nice if you give a some-what detailed account of the proof below yourself and accept that as the answer. Cheers. $\endgroup$
    – Willie Wong
    Jul 24, 2011 at 13:41

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It's called Prüfer's first theorem and I found it on page 173 in

A. G. Kurosh: The Theory of Groups, Volume One, Second English Edition. Chelsea Publishing Company, New York 1960

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