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Precobalanced subgroups of abelian groups. (English) Zbl 0717.20040

This note introduces the concept of a precobalanced subgroup of an abelian group. Specifically, a subgroup B of an abelian group A is precobalanced in A if for any subgroup K of B with B/K isomorphic to a subgroup of the rationals Q, there are subgroups \(K_ 1,...,K_ n\) of A satisfying: (1) Each \(A/K_ i\) is isomorphic to a subgroup of Q; (2) \(K=\cap \{K_ i+B|\) \(1\leq i\leq n\}\); (3) For each \(b\in B\), the height of \(b+K\) in B/K is the infimum of \(\{\) height b\(+K_ i\) in \(A/K_ i|\) \(1\leq i\leq n\}.\)
This notion dualizes that of a prebalanced subgroup as introduced by Richman and later studied by Fuchs and Viljoen. An exact sequence \(0\to B\to A\to C\to 0\) is called precobalanced (or prebalanced) whenever the embedding \(0\to B\to A\) is. Exact sequences of Butler groups (pure subgroups of finite direct sums of subgroups of Q) are always prebalanced and precobalanced. A more precise relationship is given by Corollary 6. Let E: \(0\to B\to A\to C\to 0\) be an exact sequence of torsion-free abelian groups. (a) If B is a Butler group and E is precobalanced, then E is prebalanced. (b) If C is a Butler group and E is prebalanced, then E is precobalanced.
The authors derive a number of other interesting results involving precobalanced subgroups. As a final example, we quote: Theorem 10. For any abelian groups B and C, the group of precobalanced extensions, Pcext(C,B), is a pure subgroup of Ext(C,B) that contains the torsion subgroup of Ext(C,B).
Reviewer: C.Vinsonhaler

MSC:

20K27 Subgroups of abelian groups
20K15 Torsion-free groups, finite rank
20K35 Extensions of abelian groups
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[1] DOI: 10.1112/plms/s3-15.1.680 · Zbl 0131.02501 · doi:10.1112/plms/s3-15.1.680
[2] Fuchs L., Infinite Abelian Groups 2 (1970) · Zbl 0209.05503
[3] Fuchs L., Infinite Butler groups · Zbl 0842.20045
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[6] Giovannitti A.J., Cobalanced exact sequences · Zbl 0693.20055
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[9] DOI: 10.1080/00927878808823600 · Zbl 0637.20027 · doi:10.1080/00927878808823600
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