Cancellation and absorption of lexicographic powers of totally ordered Abelian groups

Let G and H be totally ordered Abelian groups such that, for some integer k, the lexicographic powers G ^k and H ^k are isomorphic (as ordered groups). It was proved by F. Oger that G and H need not be isomorphic. We show here that they are whenever G is either divisible or ₁ -saturated (and in a few more cases). Our proof relies on a general technique which we also use to prove that G and H must be elementary equivalent as ordered groups (a fact also proved by F. Delon and F. Lucas) and isomorphic as chains. The same technique applies to the question of whether G and H should be isomorphic as groups, but, in this last case, no hint about a possible negative answer seems available.