Every abelian group of finite exponent is isomorphic to a direct sum of finite cyclic groups?

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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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. – user641 Jul 23 '11 at 20:16
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) – rainmaker Jul 24 '11 at 10:22
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. – Willie Wong Jul 24 '11 at 13:41