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OUTLINE OF PROOF. Let G be an /tr-union of pure subgroups Hnt n<o)f that are free. Write Hn= ]C*er(w) {gi}> For simplicity of notation, let JU denote the smallest ordinal having the cardinality of G. We claim that there exist subgroups Aat a<fx, of G satisfying the following conditions: (0) ^o = 0. (1) Aa is pure in G for each a<fx. (2) {AayHn} is pure in G for each a</x and each n<w. (3) Aa+i^Aa for each a such that a + Kju . (4) A a+i/A a is countable for each a such that a +1< ju. (5) AaC\Hn= Yliei(n,a){gi} for OJ</X and #<co, where I(n, a) is a subset of I(n).

On the freeness of abelian groups: A generalization of Pontryagin's theorem