Mathematische Annalen

, Volume 333, Issue 4, pp 759–785 | Cite as

On the maximal cardinality of half-factorial sets in cyclic groups

  • Alain PlagneEmail author
  • Wolfgang A. Schmid


We consider the function μ(G), introduced by W. Narkiewicz, which associates to an abelian group G the maximal cardinality of a half-factorial subset of it. In this article, we start a systematic study of this function in the case where G is a finite cyclic group and prove several results on its behaviour. In particular, we show that the order of magnitude of this function on cyclic groups is the same as the one of the number of divisors of its cardinality.

Mathematics Subject Classification (2000)

11R27 11B75 11P99 20D60 20K01 05E99 13F05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, D.D. (ed.): Factorization in integral domains, vol. 189 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1997Google Scholar
  2. 2.
    Chapman, S.T.: On the Davenport constant, the cross number and their application in factorization theory. In: Anderson, D.F., Dobbs, D.E. (eds.) Zero-dimensional commutative rings, vol. 171 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1995, pp. 167–190Google Scholar
  3. 3.
    Chapman, S.T., Coykendall, J.: Half-factorial domains, a survey. In: Chapman, S.T., Glaz, S. (eds.) Non-Noetherian commutative ring theory, vol. 520 of Math. Appl., Kluwer Acad. Publ., Dordrecht, 2000, pp. 97–115Google Scholar
  4. 4.
    Chapman, S.T., Geroldinger, A.: Krull domains and monoids, their sets of lengths, and associated combinatorial problems. In: [1], pp. 73–112Google Scholar
  5. 5.
    Chapman, S.T., Krause, U., and Oeljeklaus, E.: Monoids determined by a homogeneous linear Diophantine equation and the half-factorial property. J. Pure Appl. Algebra 151(2), 107–133 (2000)CrossRefGoogle Scholar
  6. 6.
    Chapman, S.T., Smith, W.W.: Factorization in Dedekind domains with finite class group. Israel J. Math. 71(1), 65–95 (1990)Google Scholar
  7. 7.
    Coykendall, J.: On the integral closure of a half-factorial domain. J. Pure Appl. Algebra 180(1–2), 25–34 (2003)Google Scholar
  8. 8.
    Deshouillers, J.-M., Dress, F., and Tenenbaum, G.: Lois de répartition des diviseurs. I. Acta Arith. 34(4), 273–285 (1979)Google Scholar
  9. 9.
    Erdős, P., Zaks, A.: Reducible sums and splittable sets. J. Number Theory 36(1), 89–94 (1990)CrossRefGoogle Scholar
  10. 10.
    Gao, W., Geroldinger, A.: Half-factorial domains and half-factorial subsets of abelian groups. Houston J. Math. 24(4), 593–611 (1998)Google Scholar
  11. 11.
    Geroldinger, A.: Ein quantitatives Resultat über Faktorisierungen verschiedener Länge in algebraischen Zahlkörpern. Math. Z. 205(1), 159–162 (1990)Google Scholar
  12. 12.
    Geroldinger, A., Göbel, R.: Half-factorial subsets in infinite abelian groups. Houston J. Math. 29(4), 841–858 (2003)Google Scholar
  13. 13.
    Geroldinger, A., Halter-Koch, F., and Kaczorowski, J.: Non-unique factorizations in orders of global fields. J. Reine Angew. Math. 459, 89–118 (1995)Google Scholar
  14. 14.
    Geroldinger, A., Kaczorowski, J.: Analytic and arithmetic theory of semigroups with divisor theory. Sém. Théor. Nombres Bordeaux (2) 4(2), 199–238 (1992)Google Scholar
  15. 15.
    Geroldinger, A., Schneider, R.: The cross number of finite abelian groups. II. European J. Combin. 15(4), 399–405 (1994)CrossRefGoogle Scholar
  16. 16.
    Halter-Koch, F.: Chebotarev formations and quantitative aspects of nonunique factorizations. Acta Arith. 62(2), 173–206 (1992)Google Scholar
  17. 17.
    Halter-Koch, F.: Finitely generated monoids, finitely primary monoids, and factorization properties of integral domains. In: [1], pp. 31–72Google Scholar
  18. 18.
    Halter-Koch, F.: Ideal systems. An introduction to multiplicative ideal theory, vol. 211 of Monographs and Textbooks in Pure and Applied Mathematics. Dekker, New York, 1998Google Scholar
  19. 19.
    Hassler, W.: A note on half-factorial subsets of finite cyclic groups. Far East J. Math. Sci. (FJMS) 10(2), 187–197 (2003)Google Scholar
  20. 20.
    Kaczorowski, J.: Some remarks on factorization in algebraic number fields. Acta Arith. 43(1), 53–68 (1983)Google Scholar
  21. 21.
    Kainrath, F.: On local half-factorial orders. In: Chapman, S.T. (ed.) Arithmetical properties of commutative rings and monoids, vol. 241 of Lecture Notes in Pure Appl. Math., CRC Press (Taylor & Francis Group), Boca Raton, 2005, pp. 316–324Google Scholar
  22. 22.
    Krause, U.: A characterization of algebraic number fields with cyclic class group of prime power order. Math. Z. 186(2), 143–148 (1984)CrossRefGoogle Scholar
  23. 23.
    Krause, U., Zahlten, C.: Arithmetic in Krull monoids and the cross number of divisor class groups. Mitt. Math. Ges. Hamburg 12(3), 681–696 (1991)Google Scholar
  24. 24.
    Narkiewicz, W.: Finite abelian groups and factorization problems. Colloq. Math. 42, 319–330 (1979)Google Scholar
  25. 25.
    Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers, third edition. Springer-Verlag, Berlin, 2004Google Scholar
  26. 26.
    Plagne, A., Schmid, W.A.: On large half-factorial sets in elementary p-groups: Maximal cardinality and structural characterization. Israel J. Math. 145, 285–310 (2005)Google Scholar
  27. 27.
    Radziejewski, M.: Oscillations of error terms associated with certain arithmetical functions. Monatsh. Math. 144(2), 113–130 (2004)CrossRefGoogle Scholar
  28. 28.
    Radziejewski, M.: On the distribution of algebraic numbers with prescribed factorization properties. Acta Arith. 116(2), 153–171 (2005)Google Scholar
  29. 29.
    Radziejewski, M.: The Ψ1 conjecture computations. Available online at Radziejewski's website
  30. 30.
    Skula, L.: On c-semigroups. Acta Arith. 31(3), 247–257 (1976)Google Scholar
  31. 31.
    Śliwa, J.: Factorizations of distinct lengths in algebraic number fields. Acta Arith. 31(4), 399–417 (1976)Google Scholar
  32. 32.
    Śliwa, J.: Remarks on factorizations in algebraic number fields. Colloq. Math. 46(1), 123–130 (1982)Google Scholar
  33. 33.
    Sylvester, J.J.: Mathematical Questions with their solutions, Educational Times 41, 21 (1884)Google Scholar
  34. 34.
    Tenenbaum, G.: Introduction à la théorie analytique et probabiliste des nombres, second edition, vol. 1 of Cours Spécialisés, S.M.F., Paris, 1995Google Scholar
  35. 35.
    Zaks, A.: Half factorial domains. Bull. Am. Math. Soc. 82(5), 721–723 (1976)Google Scholar
  36. 36.
    Zaks, A.: Half-factorial-domains. Israel J. Math. 37(4), 281–302 (1980)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Centre de Mathématiques Laurent SchwartzUMR 7640 du CNRSPalaiseau CedexFrance
  2. 2.Institut für Mathematik und Wissenschaftliches RechnenKarl-Franzens-Universität GrazGrazAustria

Personalised recommendations