Note on Hahn’s theorem on ordered abelian groups

In 1907, Hahn [2] showed that every (totally) ordered abelian group can be embedded in a lexicographically ordered, real function space. His proof occupies twenty-seven pages, not counting preliminaries, and may well be described as a transfinite marathon. For forty-five years, no one offered a simpler proof. But in 1952, Hausner and Wendel [3] gave a two-page proof of the same theorem for an ordered real vector space. In the present note, it is shown that the Hausner-Wendel proof applies equally well to the general case, with a few minor modifications, thus affording us an "accessible" proof of Hahn's fundamental theorem. Since this note is simply an appendix to [3 ], their definitions will not be repeated here. (Added in proof. Not until after this paper was printed did the author become aware of the fine work of Paul F. Conrad [4], who not only simplifies the proof, but extends Hahn's Theorem to partially ordered abelian groups and to even more general systems. Conrad bases his proof on the intrinsic notion of a "decomposition" of the given group G, instead of the extrinsic notion of an order isomorphism of G into an ordered function space. While basically equivalent, the two methods appear quite different, and perhaps it is just as well to have both in print.) We begin by reducing the general case to that of an ordered rational vector space. It was shown by Baer [1, p. 768], that any abelian group G having no element 5z? 0 of finite order can be embedded in a minimal "complete" abelian group V, i.e. one having the property that, for any v E V and any positive integer n, there exists x E V such that nx =v. Baer takes for V the set of all pairs (x, n), with xCG and n a positive integer, defining equality by