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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 \(k\geq 4\).

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
Full Text: DOI
[1] D.M. Arnold: Finite Rank Torsion-Free Abelian Groups and Rings. Lecture Notes 931. Springer, Berlin, 1982.
[2] D.M. Arnold: Abelian Groups and Representations of Partially Ordered Finite Sets. CMS Advanced Books in Mathematics. Springer, New York, 2000. · Zbl 0959.16011
[3] D.M. Arnold, M. Dugas: Finite rank Butler groups with small typesets. Abelian Groups and Modules (Dublin 1998). Trends in Math. (P. Eklof et al., eds.). Birkhäuser, Basel, 1998, pp. 107–119.
[4] D.M. Arnold, D. Simson: Representations of finite partially ordered sets over commutative artinian uniserial rings. J. Pure Appl. Algebra 205 (2006), 640–659. · Zbl 1106.16016 · doi:10.1016/j.jpaa.2005.07.017
[5] D.M. Arnold, D. Simson: Representations of finite posets over discrete valuation rings. Commun. Algebra 35 (2007), 3128–3144. · Zbl 1142.16003 · doi:10.1080/00927870701405173
[6] D.M. Arnold, A. Mader, O. Mutzbauer, E. Solak: Almost completely decomposable groups and unbounded representation type. J. Algebra 349 (2012), 50–62. · Zbl 1253.20057 · doi:10.1016/j.jalgebra.2011.10.019
[7] V.M. Bondarenko: Representations of bundles of semichained sets and their applications. St. Petersburg Math. J. 3 (1992), 973–996. · Zbl 0791.06002
[8] R. Burkhardt: On a special class of almost completely decomposable groups I. Abelian Groups and Modules. Proc. Udine Con. 1984, CISM Courses and Lecture 287 (R. Göbel at al., eds.). Springer, Vienna, 1984, pp. 141–150. · Zbl 0569.20044
[9] A. L. S. Corner: A note on rank and direct decompositions of torsion-free Abelian groups. Proc. Camb. Philos. Soc. 57 (1961), 230–233. · Zbl 0100.02903 · doi:10.1017/S0305004100035106
[10] J.A. Drozd: Matrix problems and categories of matrices. J. Sov. Math. 3 (1975), 692–699. · Zbl 0342.15011 · doi:10.1007/BF01084669
[11] M. Dugas: BR-groups with type set (1, 2). Forum Math. 13 (2001), 143–148. · Zbl 0964.20034 · doi:10.1515/form.2001.003
[12] T. Faticoni, P. Schultz: Direct decompositions of acd groups with primary regulating index. Abelian Groups and Modules. Proc. 1995 Colorado Springs Conference (D. Arnold et al., eds.). Marcel Dekker, New York, 1996, pp. 233–241. · Zbl 0869.20038
[13] L. Fuchs: Infinite Abelian Groups, Vol. II. Pure and Applied Mathematics 36, Academic Press, New York, 1973. · Zbl 0257.20035
[14] N. Jacobson: Basic Algebra I. W.H. Freeman and Company, San Francisco, 1974.
[15] E. L. Lady: Almost completely decomposable torsion-free Abelian groups. Proc. Am. Math. Soc. 45 (1974), 41–47. · Zbl 0292.20051 · doi:10.1090/S0002-9939-1974-0349873-6
[16] E. L. Lady: Nearly isomorphic torsion-free abelian groups. J. Algebra 35 (1975), 235–238. · Zbl 0322.20025 · doi:10.1016/0021-8693(75)90048-4
[17] A. Mader: Almost Completely Decomposable Groups. Gordon and Breach, Amsterdam, 2000. · Zbl 0945.20031
[18] A. Mader, L. Strüngmann: Generalized almost completely decomposable groups. Rend. Semin. Mat. Univ. Padova 113 (2005), 47–69. · Zbl 1149.20045
[19] O. Mutzbauer: Regulating subgroups of Butler groups. Abelian Groups. Proc. 1991 Curaçao Conf. Lecture Notes Pure Appl. Math. 146 (L. Fuchs, ed.). Marcel Dekker, New York, 1993, pp. 209–216. · Zbl 0801.20029
[20] O. Mutzbauer, E. Solak: (1, 2)-groups with p 3-regulator quotient. J. Algebra 320 (2008), 3821–3831. · Zbl 1159.20028 · doi:10.1016/j.jalgebra.2008.09.002
[21] L.A. Nazarova, A.V. Roiter: Finitely generated modules over a dyad of local Dedekind rings, and finite groups with an Abelian normal divisor of index p. Math. USSR Izv. 3 (1969), 65–89. (In Russian.) · Zbl 0207.04803 · doi:10.1070/IM1969v003n01ABEH000748
[22] L.A. Nazarova, A.V. Roiter, V.V. Sergeichuk, V.M. Bondarenko: Applications of modules over a dyad for the classification of finite p-groups possessing an Abelian subgroup of index p, and of pairs of mutually annihilating operators. J. Sov. Math. 3 (1975), 636–653; translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 28 (1972), 69–92. (In English. Russian original.) · Zbl 0346.13005 · doi:10.1007/BF01084666
[23] V.V. Sergeichuk: Canonical matrices for linear matrix problems. Linear Algebra Appl. 317 (2000), 53–102. · Zbl 0967.15007 · doi:10.1016/S0024-3795(00)00150-6
[24] H. Shapiro: A survey of canonical forms and invariants for unitary similarity. Linear Algebra Appl. 147 (1991), 101–167. · Zbl 0723.15007 · doi:10.1016/0024-3795(91)90232-L
[25] D. Simson: Linear Representations of Partially Ordered Sets and Vector Space Categories. Algebra, Logic and Applications Appl., Vol. 4. Gordon and Breach, Brooklyn, 2000.
[26] E. Solak: Almost completely decomposable groups of type (1, 2). DissertationWürzburg. (2007).
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