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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Effective bounds for the maximal order of an element in the symmetric group
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by Jean-Pierre Massias, Jean-Louis Nicolas and Guy Robin PDF
Math. Comp. 53 (1989), 665-678 Request permission


Let $\sigma _n$ be the symmetric group of n elements and \[ g(n) = \max _{\sigma \in \sigma _n}(\text {order of $\sigma $} ).\] We give here some effective bounds for $g(n)$ and $P(g(n))$ (greatest prime divisor of $g(n)$). Theoretical proofs are in "Evaluation asymptotique de l’ordre maximum d’un élément du groupe symétrique" (Acta Arith., v. 50, 1988, pp. 221-242). The tools used here are techniques of superior highly composite numbers of Ramanujan and bounds of Rosser and Schoenfeld on the Chebyshev function $\theta (x)$.
    E. Landau, "Über die Maximalordnung der Permutationen gegebenen Grades," Arch. Math. Phys. Ser. 3, v. 5, 1903, pp. 92-103.
  • Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände, Chelsea Publishing Co., New York, 1953 (German). 2d ed; With an appendix by Paul T. Bateman. MR 0068565
  • Jean-Pierre Massias, Majoration explicite de l’ordre maximum d’un élément du groupe symétrique, Ann. Fac. Sci. Toulouse Math. (5) 6 (1984), no. 3-4, 269–281 (1985) (French, with English summary). MR 799599
  • J. P. Massias, Ordre Maximum d’un Élément du Groupe Symétrique et Applications, Thèse de 3$^{\text {\`eme}}$ cycle, Université de Limoges, 1984.
  • J.-P. Massias, J.-L. Nicolas, and G. Robin, Évaluation asymptotique de l’ordre maximum d’un élément du groupe symétrique, Acta Arith. 50 (1988), no. 3, 221–242 (French). MR 960551, DOI 10.4064/aa-50-3-221-242
  • J. P. Massias & G. Robin, "Calculs effectifs sur le k$^{\text {\`eme}}$ nombre premier." (To appear.)
  • William Miller, The maximum order of an element of a finite symmetric group, Amer. Math. Monthly 94 (1987), no. 6, 497–506. MR 935414, DOI 10.2307/2322839
  • F. Morain, Tables sur la Fonction $g(n)$, Département de Math., Université de Limoges, 1988.
  • Jean-Louis Nicolas, Ordre maximal d’un élément du groupe $S_{n}$ des permutations et “highly composite numbers”, Bull. Soc. Math. France 97 (1969), 129–191 (French). MR 254130
  • Jean-Louis Nicolas, Calcul de l’ordre maximum d’un élément du groupe symétrique $S_{n}$, Rev. Francaise Informat. Recherche Opérationnelle 3 (1969), no. Sér. R-2, 43–50 (French, with Loose English summary). MR 0253514
  • J.-L. Nicolas and G. Robin, Majorations explicites pour le nombre de diviseurs de $N$, Canad. Math. Bull. 26 (1983), no. 4, 485–492 (French, with English summary). MR 716590, DOI 10.4153/CMB-1983-078-5
  • Jean-Louis Nicolas, On highly composite numbers, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp. 215–244. MR 938967
  • S. Ramanujan, Highly composite numbers [Proc. London Math. Soc. (2) 14 (1915), 347–409], Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, pp. 78–128. MR 2280858
  • Guy Robin, Estimation de la fonction de Tchebychef $\theta$ sur le $k$-ième nombre premier et grandes valeurs de la fonction $\omega (n)$ nombre de diviseurs premiers de $n$, Acta Arith. 42 (1983), no. 4, 367–389 (French). MR 736719, DOI 10.4064/aa-42-4-367-389
  • J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
  • J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$, Math. Comp. 29 (1975), 243–269. MR 457373, DOI 10.1090/S0025-5718-1975-0457373-7
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Math. Comp. 53 (1989), 665-678
  • MSC: Primary 11N45; Secondary 11Y70, 20B05, 20D60
  • DOI:
  • MathSciNet review: 979940