TITLE: Every abelian group of finite exponent is isomorphic to a direct sum of finite cyclic groups?
QUESTION [7 upvotes]: 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 $\sup_{g \in G} \mathopen|G.g\mathclose|$

REPLY [5 votes]: 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