Geodesic
A note on weakly $\aleph_1$-separable $p$-groups
Vladikavkazskij matematičeskij žurnal, Tome 9 (2007) no. 1, pp. 30-37

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It is well-known by Hill-Griffith that there exist $\aleph_1$-separable $p$-primary groups which are not direct sums of cycles. A problem of challenging interest, mainly due to Hill (Rocky Mount. J. Math., 1971), is under what extra circumstances on the group structure this holds untrue, that is every $\aleph_1$-separable $p$-group is a direct sum of cyclic groups. We prove here that any weakly $\aleph_1$-separable $p$-group of cardinality not exceeding $\aleph_1$ is quasi-complete precisely when it is a bounded direct sum of cycles, thus partly answering the posed question in the affirmative. 
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     title = {A note on weakly $\aleph_1$-separable $p$-groups},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
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P. V. Danchev. A note on weakly $\aleph_1$-separable $p$-groups. Vladikavkazskij matematičeskij žurnal, Tome 9 (2007) no. 1, pp. 30-37. http://geodesic.mathdoc.fr/item/VMJ_2007_9_1_a2/