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The largest prime dividing the maximal order of an element of \(S_ n\). (English) Zbl 0824.11059
Let \(g(n)\) denote the maximal order of an element of the symmetric group on \(n\) letters. E. Landau [Arch. Math. Phys., III. Ser. 5, 92-103 (1903; F.d.M. 34, 233)] proved that \(\log g(n)\sim (n\log n)^{1/2}\) as \(n\to \infty\). For improved estimates see J.-P. Massias, J.-L. Nicolas and G. Robin [Acta Arith. 50, 221-242 (1988; Zbl 0646.10037)].
Let \(P(g (n))\) be the largest prime divisor of \(g(n)\). J.-L. Nicolas [Acta Arith. 14, 315-332 (1968; Zbl 0179.348)] showed that \(P(g (n))\sim (n\log n)^{1/2}\) as \(n\to \infty\). J.-P. Massias, J.-L. Nicolas and G. Robin [Math. Comput. 53, 665-678 (1989; Zbl 0675.10028)] obtained that \(P(g (n))\leq 2.86 (n\log n)^{1/2}\) for \(n\geq 2\). They conjectured that \(P(g (n)) (n\log n)^{-1/2}\) achieves a maximum \((1.265 \ldots)\) for \(n\geq 5\) at \(n=215\). In the paper under review the author proves that \(P(g (n))\leq 1.328 (n\log n)^{1/2}\) for \(n\geq 5\).

MSC:
11N45 Asymptotic results on counting functions for algebraic and topological structures
20B40 Computational methods (permutation groups) (MSC2010)
20B05 General theory for finite permutation groups
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