Distribution statistique de l'ordre d'un élément du groupe symétrique. (Statistical distribution of the order of an element of the symmetric group)
Let \(S_ n\) be the symmetric group on n letters, and let \({\mathcal O}(P)\) denote the group theoretic order of P, for \(P\in S_ n\). Let \(n_ 1<n_ 2<...<n_ k\) be the different cycle-lengths in the canonical decomposition of \(P\in S_ n\). Then, \({\mathcal O}(P)=[n_ 1,n_ 2,...,n_ k].\) \textit{P. Erdős} and \textit{P. Turán} [Acta Math. Acad. Sci. Hung. 18, 309-320 (1967; Zbl 0235.20003)] proved that log \({\mathcal O}(P)\) shows an asymptotically normal distribution, namely, with the equiprobability measure on \(S_ n\), for any fixed real x, \[ (1)\quad \Pr ob(\log {\mathcal O}(P)< \log^ 2 n+x\cdot 3^{-1/2} \log^{3/2} n)= \] \[ (1+o(1))(2\pi)^{-1/2}\int^{x}_{-\infty}\exp (-t^ 2/2) dt. \] \textit{M. R. Best} [Indagationes Math. 32, 385-402 (1970; Zbl 0208.033)] and \textit{J. D. Bovey} [Bull. Lond. Math. Soc. 12, 41-46 (1980; Zbl 0436.20001)] gave new proofs for this distribution theorem. The main point of the proof of (1) is that, apart from o(n!) P's, \[ (2)\quad \exp (-3 \log n\quad (\log \log n)^ 4)\leq {\mathcal O}(P)/(n_ 1\cdot... \cdot n_ k)\leq 1. \] In the present paper the author refines (1) by improving (2) to \[ {\mathcal O}(P)/(n_ 1\cdot... \cdot n_ k)=\exp (-\log n\quad \log \log n+O(\log n\quad \log \log \log n)) \] with the exception of O(n! log\({}^{-1/2} n)\) P's. By means of the characteristic function method, he proves that \[ \Pr ob(\log {\mathcal O}(P)< \log^ 2 n-\log n\quad \log \log n+x\cdot 3^{-1/2} \log^{3/2} n)= \] \[ =(2\pi)^{- 1/2}\int^{x}_{-\infty}\exp (-t^ 2/2) dt+O(\log \log \log n\quad \log^{-1/2} n). \]
distribution of orders of elements of the symmetric group
scientific article; zbMATH DE number 3916388