Note on the extensions of Butler groups. (English) Zbl 0677.20041
A pure subgroup B of a torsion-free abelian group A is called prebalanced in A if for every a in A, the characteristic of \(a+B\) in A/B is the supremum of finitely many characteristics of \(a+b\) with b in B. The main result of this lucid paper is that if A is an extension of a Butler group B by a Butler group C, then A is Butler if and only if B is prebalanced in A. The authors study various properties of prebalanced exact sequences and link the concept to results of A. Giovannitti [Lect. Notes Math. 1006, 164-170 (1983; Zbl 0525.20040)] and the notion of decency due to U. Albrecht and P. Hill [Czech. Math. J. 37(112), 293-309 (1987; Zbl 0628.20045)].
Reviewer: P.Schultz
MSC:
20K15 | Torsion-free groups, finite rank |
20K20 | Torsion-free groups, infinite rank |
20K27 | Subgroups of abelian groups |
20K35 | Extensions of abelian groups |
Keywords:
pure subgroup; torsion-free abelian group; characteristics; extension; Butler group; prebalanced exact sequences; decencyReferences:
[1] | DOI: 10.1007/BF01304857 · Zbl 0244.20064 · doi:10.1007/BF01304857 |
[2] | DOI: 10.1007/BFb0103700 · doi:10.1007/BFb0103700 |
[3] | Albrecht, Czechoslovak Math. J. 37 pp 293– (1987) |
[4] | Arnold, Finite Rank Torsion-free Abelian Groups and Rings 931 (1982) · Zbl 0493.20034 · doi:10.1007/BFb0094245 |
[5] | Fuchs, Infinite Abelian Groups 1–2 (1970) · Zbl 0209.05503 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.