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Mathematics of Computation

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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

$\displaystyle g(n) = \max_{\sigma \in \sigma_n}($order of $ \sigma$$\displaystyle ).$

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