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On summands of direct products of Abelian groups. (English) Zbl 0538.20025
Modifying some constructions given by A. L. S. Corner [Proc. Camb. Philos. Soc. 57, 230-233 (1961; Zbl 0100.029); ibid. 66, 239-240 (1969; Zbl 0205.326)] of strange-decomposable torsion-free abelian groups, analogous results are proved for direct products of groups. Main results are (1) there exist torsion-free indecomposable groups \(\bar B\), \(\bar C\), \(E_ n\), \(rk E_ n=2\) such that \(\prod^{\infty}_{n=1}E_ n=\bar B\oplus \bar C,\) (2) there exist torsion-free indecomposable groups \(\bar B\), \(\bar C\), \(\bar D\), \(E_ n\), \(rk E_ n=1\) such that \(\prod^{\infty}_{n=1}E_ n\oplus \bar D=\bar B\oplus \bar C,\) (3) an infinite direct product of rank one torsion-free reduced groups cannot equal the direct sum of indecomposable subgroups, (4) there exists a group G such that, for every sequence of positive integers \(r_ 1,r_ 2,..\). infinitely many of which exceed 1, there exist indecomposable subgroups \(A_ n\) of rank \(r_ n\) in G such that \(G\approx \prod^{\infty}_{n=1}A_ n.\) The groups in (1), (2), (4) are obtained from the corresponding groups of Corner by taking closures in the product.
Reviewer: St.Balcerzyk
MSC:
20K20 Torsion-free groups, infinite rank
20K25 Direct sums, direct products, etc. for abelian groups
20K27 Subgroups of abelian groups
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