On the frequency of permutations containing a long cycle

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    Abstract

    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 S, 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(rs) = 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. (c) 2006 Elsevier Inc. All rights reserved.
    Original languageEnglish
    Pages (from-to)289-304
    JournalJournal of Algebra
    Volume300
    Issue number1
    DOIs
    Publication statusPublished - 2006

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