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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Almost completely decomposable torsion free abelian groups


Author: E. L. Lady
Journal: Proc. Amer. Math. Soc. 45 (1974), 41-47
MSC: Primary 20K15
MathSciNet review: 0349873
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Abstract: A finite rank torsion free abelian group $ G$ is almost completely decomposable if there exists a completely decomposable subgroup $ C$ with finite index in $ G$. The minimum of $ [G:C]$ over all completely decomposable subgroups $ C$ of $ G$ is denoted by $ i(G)$. An almost completely decomposable group $ G$ has, up to isomorphism, only finitely many summands. If $ i(G)$ is a prime power, then the rank 1 summands in any decomposition of $ G$ as a direct sum of indecomposable groups are uniquely determined. If $ G$ and $ H$ are almost completely decomposable groups, then the following statements are equivalent: (i) $ G \oplus L \approx H \oplus L$ for some finite rank torsion free abelian group $ L$. (ii) $ i(G) = i(H)$ and $ H$ contains a subgroup $ G'$ isomorphic to $ G$ such that $ [H:G']$ is finite and prime to $ i(G)$. (iii) $ G \oplus L \approx H \oplus L$ where $ L$ is isomorphic to a completely decomposable subgroup with finite index in $ G$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0349873-6
PII: S 0002-9939(1974)0349873-6
Keywords: Almost completely decomposable group, cancellation property, regulating subgroup, $ K$-equivalence
Article copyright: © Copyright 1974 American Mathematical Society