Butler's theorem revisited

Abstract. Call c.d.-group a completely decomposable Abelian group of finite rank. Butler’s Theorem proves that the class of pure subgroups of c.d. groups equals the class of torsionfree quotients of c.d.-groups. While one implication of the equivalence is proved by an applicable construction, the other is a far from applicable finite induction, and even its later proof in Arnold’s book is not constructive. We give an alternative proof of this implication by providing a workable algorithm, which minimizes the construction in certain cases, building not only the c.d. container of the Butler group but also the inclusion morphism.