TY - JOUR

T1 - On the frequency of permutations containing a long cycle

AU - Niemeyer, Alice

AU - Praeger, Cheryl

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

U2 - 10.1016/j.jalgebra.2006.02.026

DO - 10.1016/j.jalgebra.2006.02.026

M3 - Article

SN - 0021-8693

VL - 300

SP - 289

EP - 304

JO - Journal of Algebra

JF - Journal of Algebra

IS - 1

ER -