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 $\sup_{g \in G} \mathopen|G.g\mathclose|$