Abstract
Rotman [11;Corollary B] proved 1963 a certain analogon of the theorem of Jordan-Hölder for torsion-free abelian groups of finite rank defining composition sequences to be chains of pure subgroups of maximal length. For groups of rank 2 Beaumont and Pierce [2] got the complete analogon of Jordan-Hölder. It will be shown here that two composition sequences in torsion-free groups and Dedekind modules of finite rank have the same sum-type,i.e. the sum of types of composition factors. This is the complete transfer of the theorem of Jordan-Hölder.
Research supported by grant Mu 628/1–1 from the Deutsche Forschungsgemeinschaft
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References
D. Arnold, Finite rank torsion free abelian groups and rings, Lecture Notes 931 (1982).
D. Arnold, Pure subgroups of finite rank completely decomposable groups, Procedings of Abelian Group Theory (Oberwolfach), Lecture Notes 874 (1981), 1–31.
R.A. Beaumont and R.S. Pierce, Torsion free groups of rank two, Mem. Amer. Math. Soc. 38 (1961).
L. Fuchs, Infinite abelian groups I + II, New York ( 1970, 1973 ).
N. Jacobson, Basic Algebra I, San Francisco (1975).
I. Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327–340.
J. Koehler, The type set of a torsion-free group of finite rank, Ill. J. Math. 9 (1965), 66–86.
G. Kolettis, Homogeneously decomposable modules, Studies on Abelian Groups. 223–238 (Paris, 1968 ).
H. Lausch, Tensorprodukte torsionsfreier abelscher Gruppen endlichen Ranges, Dissertation (Würzburg, 1982 ).
O. Mutzbauer, Untergruppen und Faktoren torsionsfreier abelscher Gruppen des Ranges 2, Publ. Math. Debrecon, 26 (1979), 95–104.
P. Schultz, The typeset and cotypeset of a rank 2 abelian group, Pac.J.Math. 78 (1978), 503–517.
J. Rotman, The Grothendieck group of torsion-free abelian groups of finite rank, Proc. Lond. Math. Soc. (3), 13 (1963), 724–732
R. Burkhardt, to apprear.
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© 1983 Springer-Verlag Berlin Heidelberg
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Mutzbauer, O. (1983). Type Graph. In: Göbel, R., Lady, L., Mader, A. (eds) Abelian Group Theory. Lecture Notes in Mathematics, vol 1006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21560-9_10
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DOI: https://doi.org/10.1007/978-3-662-21560-9_10
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