TY - JOUR

T1 - Permutations with restricted cycle structure and an algorithmic application

AU - Beals, R.

AU - Leedham-Green, C.R.

AU - Niemeyer, Alice

AU - Praeger, Cheryl

AU - Seress, Akos

PY - 2002

Y1 - 2002

N2 - Let q be an integer with q greater than or equal to 2. We give a new proof of a result of Erdos and Turan determining the proportion of elements of the finite symmetric group S-n having no cycle of length a multiple of q. We then extend our methods to the more difficult case of obtaining the proportion of such elements in the finite alternating group A(n). In both cases, we derive an asymptotic formula with error term for the above mentioned proportion, which contains an unexpected occurrence of the Gamma-function.We apply these results to estimate the proportion of elements of order 2f in S-n, and of order 3f in A(n) and S-n, where gcd(2,f) = 1, and gcd(3,f) = 1, respectively, and log f is polylogarithmic in n. We also give estimates for the probability that the f th power of such elements is a transposition or a 3-cycle, respectively. An algorithmic application of these results to computing in A(n) or S-n, given as a black-box group with an order oracle, is discussed.

AB - Let q be an integer with q greater than or equal to 2. We give a new proof of a result of Erdos and Turan determining the proportion of elements of the finite symmetric group S-n having no cycle of length a multiple of q. We then extend our methods to the more difficult case of obtaining the proportion of such elements in the finite alternating group A(n). In both cases, we derive an asymptotic formula with error term for the above mentioned proportion, which contains an unexpected occurrence of the Gamma-function.We apply these results to estimate the proportion of elements of order 2f in S-n, and of order 3f in A(n) and S-n, where gcd(2,f) = 1, and gcd(3,f) = 1, respectively, and log f is polylogarithmic in n. We also give estimates for the probability that the f th power of such elements is a transposition or a 3-cycle, respectively. An algorithmic application of these results to computing in A(n) or S-n, given as a black-box group with an order oracle, is discussed.

U2 - 10.1017/S0963548302005217

DO - 10.1017/S0963548302005217

M3 - Article

SN - 0963-5483

VL - 11

SP - 447

EP - 464

JO - Combinatorics Probability & Computing

JF - Combinatorics Probability & Computing

IS - 5

ER -