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Type Graph

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1006)

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.

Keywords

  • Type Sequence
  • Direct Summand
  • Composition Sequence
  • Finite Rank
  • Composition Factor

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Research supported by grant Mu 628/1–1 from the Deutsche Forschungsgemeinschaft

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References

  1. D. Arnold, Finite rank torsion free abelian groups and rings, Lecture Notes 931 (1982).

    Google Scholar 

  2. D. Arnold, Pure subgroups of finite rank completely decomposable groups, Procedings of Abelian Group Theory (Oberwolfach), Lecture Notes 874 (1981), 1–31.

    Google Scholar 

  3. R.A. Beaumont and R.S. Pierce, Torsion free groups of rank two, Mem. Amer. Math. Soc. 38 (1961).

    Google Scholar 

  4. L. Fuchs, Infinite abelian groups I + II, New York ( 1970, 1973 ).

    Google Scholar 

  5. N. Jacobson, Basic Algebra I, San Francisco (1975).

    Google Scholar 

  6. I. Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327–340.

    CrossRef  Google Scholar 

  7. J. Koehler, The type set of a torsion-free group of finite rank, Ill. J. Math. 9 (1965), 66–86.

    Google Scholar 

  8. G. Kolettis, Homogeneously decomposable modules, Studies on Abelian Groups. 223–238 (Paris, 1968 ).

    Google Scholar 

  9. H. Lausch, Tensorprodukte torsionsfreier abelscher Gruppen endlichen Ranges, Dissertation (Würzburg, 1982 ).

    Google Scholar 

  10. O. Mutzbauer, Untergruppen und Faktoren torsionsfreier abelscher Gruppen des Ranges 2, Publ. Math. Debrecon, 26 (1979), 95–104.

    Google Scholar 

  11. P. Schultz, The typeset and cotypeset of a rank 2 abelian group, Pac.J.Math. 78 (1978), 503–517.

    CrossRef  Google Scholar 

  12. J. Rotman, The Grothendieck group of torsion-free abelian groups of finite rank, Proc. Lond. Math. Soc. (3), 13 (1963), 724–732

    CrossRef  Google Scholar 

  13. R. Burkhardt, to apprear.

    Google Scholar 

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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

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12335-4

  • Online ISBN: 978-3-662-21560-9

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