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Quasi-isomorphism and \(\mathbb{Z}_{(2)}\)-representations for a class of Butler groups. (English) Zbl 1147.20314

Summary: A \(\mathcal B^{(1)}\)-group is a finite rank torsion-free Abelian group which is a homomorphic image, with rank 1 kernel, of a completely decomposable group. The study of these groups reduces to that of the special form \(G=G[A_1,\dots,A_n]\), which is the cokernel of the diagonal imbedding of \(A_1\cap\cdots\cap A_n\) into \(A_1\oplus\cdots\oplus A_n\), where the \(A_i\)’s form an \(n\)-tuple of nonzero subgroups of the additive group of rational numbers. With any such group \(G\) we associate in a canonical manner a vector space representation \(\mathcal R_G\), over the 2 element field \(\mathbb{Z}_{(2)}\), of the poset \(\text{typeset}(G)\), consisting of the types realized by the nonzero elements of \(G\). Let \(H=G[B_1,\dots,B_n]\) be another such group with the same typeset. We prove that \(G\) and \(H\) are quasi-isomorphic if and only if \(\mathcal R_G\) and \(\mathcal R_H\) are isomorphic representations.

MSC:

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

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