Torsion free groups

Hill, Paul; Megibben, Charles

doi:10.1090/S0002-9947-1986-0833706-8

Torsion free groups
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by Paul Hill and Charles Megibben

Trans. Amer. Math. Soc. 295 (1986), 735-751

DOI: https://doi.org/10.1090/S0002-9947-1986-0833706-8

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Abstract:

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 $\ast$-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 ${\aleph _1}$; that is, separable groups of cardinality ${\aleph _1}$ have balanced projective dimension $\leq 1$.

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