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 |
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