Cancellation and absorption of lexicographic powers of totally ordered Abelian groups (Q1115465)

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 \textit{F. Oger} [Bull. Lond. Math. Soc. 20, 198-202 (1988; Zbl 0619.06012)] that G and H in general need not be isomorphic. The author provides an interesting general technique which shows that G and H are isomorphic whenever G is either divisible or \(\omega_ 1\)-saturated (and in a few more cases); morevoer, G and H are always isomorphic as chains and elementarily equivalent as ordered groups (this latter fact has also been proved by F. Delon and F. Lucas).