The order of elements in Sylow \(p\)-subgroups of the symmetric group. (Q812792)

In a sequence of papers, \textit{P. Erdős} and \textit{P. Turán} developed a statistical theory of the symmetric group \(S_n\) on \(n\) letters. They proved that almost all elements of \(S_n\) have order \(\exp((1/2+o(1))\log^2n)\) [Z. Wahrscheinlichkeitstheor. Verw. Geb. 4, 175-186 (1965; Zbl 0137.25602)], whereas the elements of almost all conjugacy classes of \(S_n\) have order \(\exp((c+o(1))n^{1/2})\) with a positive constant \(c\) [Period. Math. Hung. 2, 149-163 (1972; Zbl 0247.20008)]. P. Turán posed the problem of developing a statistical theory for subgroups of \(S_n\), in particular for Sylow subgroups. For a fixed prime \(p\), let \(P_n\) be a Sylow \(p\)-subgroup of \(S_{p^n}\). \textit{P. P. Pálfy} and the reviewer [in Studies in pure mathematics, Mem. of P. Turán, 531-542 (1983; Zbl 0521.20054)] proved that, for \(\omega(n)\nearrow\infty\), almost all elements of \(P_n\) have order \(\exp(M_n+O(\omega(n))\log p)\) where \(M_n\) is the mean value of a suitable random variable and \(c_1(p)n<M_n<c_2(p)n\) for sufficiently large \(n\). \textit{M. Abért} and \textit{B. Virág} [J. Am. Math. Soc. 18, No. 1, 157-192 (2005; Zbl 1135.20015)] proved that \(M_n=(1+o(1))c_2(p)n\). In the paper under review, the author defines a random variable \(\xi_n\) to be the logarithm of the common order the elements (to the base \(p\)) in a randomly chosen conjugacy class of \(P_n\) and proves \(\xi_n\) has bounded variance and mean value \((\log n)/(\log p)+O(1)\).