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Torsion Free Groups

In this paper we introduce the class of torsion free k-groups and the notion of a knice subgroup. Torsion free k-groups form a class of groups more extensive than the separable groups of Baer, but they enjoy many of the same closure properties. We establish a role for knice subgroups of torsion... Full description

1st Person: Hill, Paul
Additional Persons: Megibben, Charles verfasserin
Source: in Transactions of the American Mathematical Society Vol. 295, No. 2 (1986), p. 735-751
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Type of Publication: Article
Language: English
Published: 1986
Keywords: research-article
Online: Volltext
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003 DE-601
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008 150325s1986 000 0 eng d
024 7 |a 10.2307/2000061  |2 doi 
024 8 |a 2000061 
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040 |b ger  |c GBVCP 
041 0 |a eng 
084 |a 20K15  |2 MSC 
084 |a 20K20  |2 MSC 
084 |a Torsion free group  |2 MSC 
084 |a primitive element  |2 MSC 
084 |a *-valuated coproduct  |2 MSC 
084 |a free *-valuated subgroup  |2 MSC 
084 |a k-group  |2 MSC 
084 |a separable group  |2 MSC 
084 |a knice subgroup  |2 MSC 
084 |a completely decomposable  |2 MSC 
084 |a third axiom of countability  |2 MSC 
084 |a balanced projective dimension  |2 MSC 
100 1 |a Hill, Paul 
245 1 0 |a Torsion Free Groups  |h Elektronische Ressource 
300 |a Online-Ressource 
500 |a Copyright: Copyright 1986 American Mathematical Society 
520 |a In this paper we introduce the class of torsion free k-groups and the notion of a knice subgroup. Torsion free k-groups form a class of groups more extensive than the separable groups of Baer, but they enjoy many of the same closure properties. We establish a role for knice subgroups of torsion free groups analogous to that played by nice subgroups in the study of torsion groups. For example, among the torsion free groups, the balanced projectives are characterized by the fact that they satisfy the third axiom of countability with respect to knice subgroups. Separable groups are characterized as those torsion free k-groups with the property that all finite rank, pure knice subgroups are direct summands. The introduction of these new classes of groups and subgroups is based on a preliminary study of the interplay between primitive elements and *-valuated coproducts. As a by-product of our investigation, new proofs are obtained for many classical results on separable groups. Our techniques lead naturally to the discovery that a balanced subgroup of a completely decomposable group is itself completely decomposable provided the corresponding quotient is a separable group of cardinality not exceeding ℵ1; that is, separable groups of cardinality ℵ1have balanced projective dimension ≤ 1. 
653 |a research-article 
700 1 |a Megibben, Charles  |e verfasserin  |4 aut 
773 0 8 |i in  |t Transactions of the American Mathematical Society  |d Providence, RI : Soc  |g Vol. 295, No. 2 (1986), p. 735-751  |q 295:2<735-751  |w (DE-601)JST086523945  |x 0002-9947 
856 4 1 |u https://www.jstor.org/stable/2000061  |3 Volltext 
856 4 1 |u http://dx.doi.org/10.2307/2000061  |3 Volltext 
912 |a GBV_JSTOR 
951 |a AR 
952 |d 295  |j 1986  |e 2  |h 735-751 

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