On some problems of a statistical group theory. V (Q2547901)

[Part IV, Acta math. Acad. Sci. Hungar. 19, 413-435 (1968; Zbl 0235.20004).] Let \(S_n\) be the symmetric group of \(n\) elements. It is well known that the number of conjugacy classes of \(S_n\) is \(p(n)\) the number of partitions of \(n\). Let \(H\) be an element of \(S_n\) \(O(H)\) its orders which only depends on the conjugacy class of \(H\). \(P(H)\) denotes the greatest prime factor of \(O(H)\). The authors prove the following theorem: For almost all \(H\) (i.e. for all \(H\) except for \(o(p(n))\) of them) we have \[ |P(H)-\left({\sqrt{6n} \over 2 \pi} \log n-{\sqrt{6n} \over \pi} \log\log n \right)| < \omega (n) \sqrt n \] where \(\omega(n)\) tends to infinity as slowly as we please. [See also the authors, Acta. Math. Acad. Sci. Hung. 18, 151-163 (1967; Zbl 0189.31302).]