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

 

A note on torsion-free abelian groups of finite rank


Authors: W. Wickless and C. Vinsonhaler
Journal: Proc. Amer. Math. Soc. 39 (1973), 63-68
MSC: Primary 20K15
MathSciNet review: 0313419
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Abstract: Let $ G$ be a torsion-free abelian group of rank $ n$ and $ X = \{ {x_1}, \cdots ,{x_n}\} $ a maximal set of rationally independent elements in $ G$. It is well known that any $ g \in G$ can be uniquely written $ g = {\alpha _1}{x_1} + \cdots + {\alpha _n}{x_n}$, for some $ {\alpha _1}, \cdots ,{\alpha _n} \in Q$, the rational numbers. This enables us to define, for any such $ (G,X)$, a collection of sub-groups of $ Q$ and ``natural'' isomorphisms, denoted by $ S(G,X)$. It is known that if $ G$ is of rank two, then $ G$ may be recovered from $ S(G,X)$ is a natural way. The following result is obtained for groups of rank greater than two:

Theorem. Let $ G,G'$ be torsion free abelian groups of finite rank with $ S(G,X) = S(G',X')$ for suitable $ X,X'$. Let $ F,F'$ be the free subgroups of $ G,G'$ generated by $ X,X'$. Then $ G/F \cong G'/F'$.

An additional condition is given for pairs ( $ (G,X),(G',X')$ such that $ S(G,X) = S(G',X')$ implies $ G \cong G'$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0313419-8
PII: S 0002-9939(1973)0313419-8
Article copyright: © Copyright 1973 American Mathematical Society