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Effective bounds for the maximal order of an element in the symmetric group

Authors: Jean-Pierre Massias, Jean-Louis Nicolas and Guy Robin
Journal: Math. Comp. 53 (1989), 665-678
MSC: Primary 11N45; Secondary 11Y70, 20B05, 20D60
MathSciNet review: 979940
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Abstract: 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 $\theta (x)$.

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Keywords: Symmetric group, arithmetic function, highly composite numbers
Article copyright: © Copyright 1989 American Mathematical Society