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Indecomposable \((1,3)\)-groups and a matrix problem. (English) Zbl 1281.20063
Summary: Almost completely decomposable groups with a critical typeset of type \((1,3)\) and a \(p\)-primary regulator quotient are studied. It is shown that there are, depending on the exponent of the regulator quotient \(p^k\), either no indecomposables if \(k\leq 2\); only six near isomorphism types of indecomposables if \(k=3\); and indecomposables of arbitrary large rank if \(k\geq 4\).

MSC:
20K15 Torsion-free groups, finite rank
20K25 Direct sums, direct products, etc. for abelian groups
20K35 Extensions of abelian groups
15A21 Canonical forms, reductions, classification
16G20 Representations of quivers and partially ordered sets
20K27 Subgroups of abelian groups
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