On some problems of a statistical group theory. VII (Q2555913)

[Part V in Periodica math. Hungar. 1, 5-13 (1971; Zbl 0223.10005); Part VI in J. Indian math. Soc., n. Ser. 34 (1970), 175-192 (1971; Zbl 0235.10008)]. Denote by \(S_n\) the symmetric group of \(n\) letters. It is well known that \(S_n\) has \(p(n)\) conjugacy classes where \(p(n)\) is the number of unrestricted partitions of \(n\). The authors prove that for all but \(O(p(n))\) of these conjugacy classes the order \(O(K)\) of the elements in the conjugacy class satisfies the inequality \[ \exp\{(A_0- \epsilon) \sqrt n\} < O(K) < \exp\{(A_0+\epsilon) \sqrt n\} \] where \(A_0={2 \sqrt 6 \over \pi} \sum_{j \neq 0}{(-1)^{j+1} \over 3j^2+j}.\)