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Mixed groups. (English) Zbl 0798.20050

A direct summand of a simply presented mixed abelian group is called a (global) Warfield group. Warfield groups were extensively studied in the earlier works of R. Hunter, F. Richman and E. A. Walker leading to satisfactory classification theorems [see e.g. Trans. Am. Math. Soc. 235, 345-362 (1978; Zbl 0368.20034) and ibid. 266, 555-572 (1981; Zbl 0471.20038)].
In this paper the authors expand the concept of a knice subgroup which was successfully used in their earlier study of \(p\)-local Warfield groups [ibid. 295, 715-734 (1986; Zbl 0597.20048)]. An appropriate formulation of the primitivity of an element enables the authors to define the concept of a knice subgroup of a mixed abelian group. The main theorem characterizes a Warfield group as a mixed group possessing an axiom-3 family of knice subgroups.
Here a family \(\mathcal C\) of subgroups of a group \(G\) is said to be an axiom-3 family if \(\mathcal C\) contains \(G\) and \(\{0\}\), is closed with respect to the formation subgroup-union and if \(A\in {\mathcal C}\) and \(X\) is a countable subset of \(G\), then there is a \(B \in {\mathcal C}\) containing both \(A\) and \(X\) such that \(B/A\) is countable. A difficult earlier theorem that Warfield groups are closed under direct summands now follows without much effort from this axiom-3 characterization. As another application, the authors establish that an isotype knice subgroup of index at most \(\aleph_ 1\) in a Warfield group is again a Warfield group.

MSC:

20K21 Mixed groups
20K27 Subgroups of abelian groups
20K25 Direct sums, direct products, etc. for abelian groups
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