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Solution of a Problem of L. Fuchs Concerning Finite Intersections of Pure Subgroups

L. Fuchs states in his book “Infinite Abelian Groups” [6, Vol. I, p. 134] the following Problem 13. Find conditions on a subgroup of A to be the intersection of a finite number of pure (p-pure) subgroups of A. The answer to this problem will be given as a special case of our theorem below. In order to find a better setting of this problem recall that a subgroup S ⊆ E is p-pure if pnE ∩ S = pnS for all natural numbers. Then S is pure in E if S is p-pure for all primes p. This generalizes to pσ -isotype, a definition due to L. J. Kulikov, cf. [6, Vol. II, p. 75] and [11, pp. 61, 62]. If α is an ordinal, then S is pσ -isotype if