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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Mixed groups


Authors: Paul Hill and Charles Megibben
Journal: Trans. Amer. Math. Soc. 334 (1992), 121-142
MSC: Primary 20K21; Secondary 20K27
DOI: https://doi.org/10.1090/S0002-9947-1992-1116315-8
MathSciNet review: 1116315
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Abstract: As the culmination of a series of several papers, we establish here a combinatorial characterization of Warfield groups (that is, direct summands of simply presented abelian groups) in terms of knice subgroups--a refinement of the concept of nice subgroup appropriate to the study of groups containing elements of infinite order. Central to this theory is the class of $ k$-groups, those in which 0 is a knice subgroup, and the proof that this class is closed under the formation of knice isotype subgroups. In particular, a direct summand of a $ k$-group is a $ k$-group. As an application of our Axiom $ 3$ characterization of Warfield groups, we prove that $ k$-groups of cardinality $ {\aleph _1}$ have sequentially pure projective dimension $ \leq 1$; or equivalently, if $ H$ is a knice isotype sub-group of the Warfield group $ G$ with $ \vert G/H\vert = {\aleph _1}$, then $ H$ is itself a Warfield group.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1116315-8
Keywords: Mixed groups, Axiom $ 3$ characterization, Warfield groups, simply presented groups, knice subgroups, $ k$-groups, primitive element, decomposition basis, sequential purity
Article copyright: © Copyright 1992 American Mathematical Society