A Class of Torsion-Free Abelian Groups of Finite Rank

Butler, M. C. R.

doi:10.1112/plms/s3-15.1.680

Cited by 81 publications

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“…The class of completely decomposable torsion-free abelian groups is one of the few classes of abelian groups that can be completely determined by isomorphism invariants, as was shown by R. Baer [5]. In 1965 M. C. R. Butler [10] initiated the study of the pure subgroups of finite rank completely decomposable groups and in recent years this class of groups has been studied in greater detail by D. Arnold [2,4], D. Arnold and C. Vinsonhaler [3], L. Bican [6,7], L. Bican and L.…”

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confidence: 98%

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Infinite Rank Butler Groups

Dugas¹,

Rangaswamy²

1988

Transactions of the American Mathematical Society

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Abstract. A torsion-free abelian group G is said to be a Butler group if Bext(C, T) = 0 for all torsion groups T. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group G satisfies the T.E.P. over a pure subgroup H if and only if H is decent in G in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called B2-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a ^-group. We show under ( V = L) that this is indeed the case for Butler groups of rank Nt. On the other hand it is shown that, under ZFC, it is undecidable whether a group B for which Bext( B, T) = 0 for all countable torsion groups T is indeed a B2-group.

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confidence: 98%

“…Then A is a rank two homogeneous group of type (0,0,..., 0,... ) and is indecomposable. Since, by [10], a homogeneous Butler group must be completely decomposable, A is not a Butler group. However, A has the T.E.P.…”

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confidence: 99%

Infinite Rank Butler Groups

Dugas¹,

Rangaswamy²

1988

Transactions of the American Mathematical Society

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“…Theorem 3 in [4] asserts that a Butler group whose extractable typeset is a singleton is completely decomposable. In [6] we proved an analogue of this theorem which states that a balanced Butler group with extractable typeset of cardinality at most two is completely decomposable.…”

Section: If T(h) Contains Two Distinct Types T1 and T2 Such That Hh) mentioning

confidence: 99%

“…There are a number of similarities between these results and some published results on pure subgroups of finite rank completely decomposable groups and we employ some techniques developed by Butler in [4] and Arnold in [1]. Specifically, we show that if a finite rank completely decomposable group has an extractable typeset of cardinality at most five, then all its balanced subgroups are completely decomposable.…”

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confidence: 99%

Balanced Subgroups of Finite Rank Completely Decomposable Abelian Groups

Nongxa¹

1987

Transactions of the American Mathematical Society

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ABSTRACT. It is proved that, if a finite rank completely decomposable group has extractable typeset of cardinality at most 5, all its balanced subgroups are also completely decomposable. Balanced Butler groups with extractable typeset of size at most 3 are almost completely decomposable and decompose into rank 1 and/or rank 3 indecomposable summands. We also construct an indecomposable balanced Butler group whose extractable typeset is of size 4 which fails to be almost completely decomposable.In this note we wish to establish a few results on the structure of balanced subgroups of finite rank completely decomposable groups. There are a number of similarities between these results and some published results on pure subgroups of finite rank completely decomposable groups and we employ some techniques developed by Butler in [4] and Arnold in [1]. Specifically, we show that if a finite rank completely decomposable group has an extractable typeset of cardinality at most five, then all its balanced subgroups are completely decomposable. In [6] we showed that there exists a finite rank completely decomposable group with extractable typeset of size six which contains an indecomposable balanced subgroup. We also showed that if H is a balanced subgroup of a finite rank completely decomposable group and the extractable typeset of H contains at most two elements, then H is completely decomposable. We give an alternative proof of this theorem. We also show that if H is balanced in a finite rank completely decomposable group and the extractable typeset of H is of size three, then H is almost completely decomposable and decomposes into rank 1 and/or rank 3 indecomposable summands. We construct an indecomposable balanced subgroup H of a finite rank completely decomposable group, such that the extractable typset of H is of size 4 and H fails to be almost completely decomposable.All the groups we consider here are assumed to be abelian and for general notation, terminology, and results we refer the reader to [5].Let G be a torsion-free group and let 9 E G. We shall denote by Xc(g), the height-sequence or characteristic of 9 in G and typeC (g) will mean the type of 9 in G.T(G) = {typec(g): 0"# 9 E G} will be called the typeset of G. If 8 = {Xl, X2, ... , Xn} is a finite set of heightsequences then inf(8) and sup(8) are height-sequences given by component minimums and maximums of the height-sequences in 8. If 8' = {71' 72, ... ,7 n} is a finite set of types with Xi E 7i, 1 :::; i :::; n, then inf(8') and sup(8') are the types

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“…0 -group (called torsionless in [6]) if G*(T) is a pure subgroup of G for each type T, where G*(r) is the subgroup of G generated by {x e G|type G (x) > T}. Among the class of almost completely decomposable groups, the 2?…”

Section: (B) the Group G May Be Chosen With Rank = 2mrank(r) Where Mmentioning

confidence: 99%