zbMATH — the first resource for mathematics

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
Full Text: