An isomorphic refinement theorem for Abelian groups

Lemma 1 is obvious except perhaps for property (ii) for which see Lemma 2 is immediate from the finite exchange property. The groups in ^ actually have the exchange property (not just the finite exchange property). For a proof we refer to [3, Lemma 3.6], only remarking that one must use our Lemmas 6 and 7 elow instead of Crawley's 3.2 and 3.3. We will not need this result.