Mixed groups

Hill, Paul; Megibben, Charles

doi:10.1090/S0002-9947-1992-1116315-8

Mixed groups
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Trans. Amer. Math. Soc. 334 (1992), 121-142 Request permission

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 $|G/H| = {\aleph _1}$, then $H$ is itself a Warfield group.

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Simply presented groups