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Quasi-isomorphism invariants for a class of torsion-free Abelian groups. (English) Zbl 0695.20031

The paper is concerned with classifying special Butler groups of the following kind. Let \(A_ i\), \(i=1,...,n\), be subgroups of \({\mathbb{Q}}\) and let \(G=G(A_ 1,...,A_ n)\) be the kernel of the summation mapping \(A_ 1\oplus...\oplus A_ n\to {\mathbb{Q}}:\) \((a_ i)\mapsto \sum a_ i\). F. Richman [Trans. Am. Math. Soc. 279, 175-185 (1983; Zbl 0524.20028)] showed that the types of the \(A_ i\) form a complete set of quasi- isomorphism invariants for G provided that each projection \(G\to A_ i\) is surjective and that the types \(type(A_ i\cap A_ j)\), \(1\leq i<j\leq n\), are pairwise incomparable. The authors introduce invariants \(r_ G(\tau,\sigma)=rk((G(\tau)+G[\sigma])/G[\sigma])\) where \(\tau\) and \(\sigma\) are types, \(G(\tau)=\{x\in G:\) type \(x\geq \tau \}\), \(G[\sigma]=\cap \{Ker f:\) \(f\in Hom(G,X_{\sigma})\}\), and \(X_{\sigma}\) is a rank-1 group of type \(\sigma\). From these invariants the types of the \(A_ i\) can be recovered. It is shown that the ranks \(r_ G(\tau,\sigma)\) form a complete set of quasi-isomorphism invariants for a class of groups of the sort \(G(A_ 1,...,A_ n)\), called CT- groups. Another major result is that the class of finite direct sums of CT-groups is closed under direct summands. The major tool are the so- called representing graphs of Butler groups, in fact, CT-groups are those which possess a representing graph with pairwise incomparable labels of edges. Unfortunately, the definition of representing graphs is a bit technical and too long to reproduce here.
Reviewer: A.Mader

MSC:

20K15 Torsion-free groups, finite rank
20K25 Direct sums, direct products, etc. for abelian groups

Citations:

Zbl 0524.20028
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