On the frequency of permutations containing a long cycle

A general explicit upper bound is obtained for the proportion P(n,m) of elements of order dividing m, where n−1⩽m⩽cn for some constant c, in the finite symmetric group Sn. This is used to find lower bounds for the conditional probabilities that an element of Sn or An contains an r-cycle, given that it satisfies an equation of the form xrs=1 where s⩽3. For example, the conditional probability that an element x is an n-cycle, given that xn=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.