ON SOME PROBLEMS OF A STATISTICAL GROUP THEORY, VI By P. ERDOS and P. TURAN

(To the memory of S. Minakshisundaram) 1. In the second paper of this series we proved the following two theorems. Let Sn stand for the symmetric group with n letters, P a generic element of it and O(P) its order. Then we have THEOREM A. For almost all P's in Sn , i .e. with the exception of o(n!) P's at most, 0(P) is divisible by all prime powers not exceeding log n 1 + 3 log log log n w(n) log log n log log n log log n if only w (n) X oc arbitrarily slowly. The other theorem shows that the theorem is best possible in the following strong sense. THEOREM B. If w (n) >r + oo arbitrarily slowly, then almost no P's (i .e. only o(n!) of it) have the property that 0(P) is divisible by all primes not exceeding log n log log log n w(n) log log n I 1+3 log log n + log log n Since the P's in a conjugacy class H of Sn have the same order, we may denote by O(H) the common order of its elements and it is natural to ask the corresponding statistical theorem for O(H). The total number of conjugacy classes in Sn is, as well known, p(n), the number of partitions of n. As announced in the fifth paper