This article is cited in 3 papers

A note on weakly $\aleph_1$-separable $p$-groups

P. V. Danchev

Plovdiv State University «Paissii Hilendarski», Plovdiv, Bulgaria

Abstract: 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.

Key words: weakly $\aleph_1$-separable groups, quasi-complete groups, torsion-complete groups, bounded groups.

UDC: 512.742

MSC: 20K 10

Received: 03.07.2006

Language: English

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