On the Frequency of Permutations Containing a Long Cycle
نویسندگان
چکیده
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.
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