Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T00:27:15.039Z Has data issue: false hasContentIssue false
  • English
  • Français

Permutations with Restricted Cycle Structure and an Algorithmic Application

Published online by Cambridge University Press:  09 October 2002

ROBERT BEALS ,
CHARLES R. LEEDHAM-GREEN ,
ALICE C. NIEMEYER ,
CHERYL E. PRAEGER  and
ÁKOS SERESS
ROBERT BEALS
Affiliation:
IDA Center for Communications Research, Princeton, NJ 08540, USA (e-mail: beals@idaccr.org)
CHARLES R. LEEDHAM-GREEN
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, England (e-mail: crlg@maths.qmw.ac.uk)
ALICE C. NIEMEYER
Affiliation:
Department of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia (e-mail: alice@maths.uwa.edu.au)
CHERYL E. PRAEGER
Affiliation:
Department of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia (e-mail: alice@maths.uwa.edu.au)
ÁKOS SERESS
Affiliation:
Department of Mathematics, The Ohio State University, Columbus OH 43210, USA (e-mail: akos@math.ohio-state.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let q be an integer with q [ges ] 2. We give a new proof of a result of Erdös and Turán determining the proportion of elements of the finite symmetric group Sn 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 An. 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 Sn, and of order 3f in An and Sn, 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 fth power of such elements is a transposition or a 3-cycle, respectively. An algorithmic application of these results to computing in An or Sn, given as a black-box group with an order oracle, is discussed.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 11 , Issue 5 , September 2002 , pp. 447 - 464
Copyright
2002 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)