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Journal of the London Mathematical Society Advance Access originally published online on November 12, 2007
Journal of the London Mathematical Society 2007 76(3):622-632; doi:10.1112/jlms/jdm084
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© 2007 London Mathematical Society

On the proportion of permutations of order a multiple of the degree

Alice C. Niemeyer and Cheryl E. Praeger

School of Mathematics & Statistics
University of Western Australia
35 Stirling Highway
Crawley
WA 6009
Australia
alice{at}maths.uwa.edu.au

We study permutations of a set of size n for which the order is a multiple of n. We prove that, for large n, most such elements lie in one of two families. The first family consists of those permutations with a single very large cycle of order dividing n and includes the n-cycles, and the second consists of permutations for which the cycles of length dividing n have total length significantly less than n. This work was inspired by the algorithmic problem of fast recognition of large symmetric groups acting primitively on subsets.

2000 Mathematics Subject Classification 20B30 (primary); 20P05 (secondary).

Received March 31, 2005; revised December 7, 2006; published online November 12, 2007.


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