On the frequency of permutations containing a long cycle

Abstract

A general explicit upper bound is obtained for the proportion P ( n , m ) of elements of order dividing m, where n − 1 ⩽ m ⩽ c n for some constant c, in the finite symmetric group S n . This is used to find lower bounds for the conditional probabilities that an element of S n or A n contains an r-cycle, given that it satisfies an equation of the form x r s = 1 where s ⩽ 3 . For example, the conditional probability that an element x is an n-cycle, given that x n = 1 , is always greater than 2/7, and is greater than 1/2 if n does not divide 24. Our results improve estimates of these conditional probabilities in earlier work of the authors with Beals, Leedham-Green and Seress, and have applications for analysing black-box recognition algorithms for the finite symmetric and alternating groups.